The Statistics of DNA Capture and Re-Capture by a Solid-State Nanopore by Mirna Mihovilovic B. Sc., University of Zagreb, Zagreb, Croatia, 2008 A Dissertation submitted in partial fulllment of the requirements for the Degree of Doctor of Philosophy in the Department of Physics at Brown University PROVIDENCE, RHODE ISLAND May 2014 c Copyright 2014 by Mirna Mihovilovic This dissertation by Mirna Mihovilovic is accepted in its present form by the Department of Physics as satisfying the dissertation requirement for the degree of Doctor of Philosophy. Date prof. Derek Stein, Advisor Recommended to the Graduate Council Date prof. James Valles, jr., Reader Date prof. See-Chen Ying, Reader Approved by the Graduate Council Date prof. Peter M. Weber, Dean of the Graduate School iii Vitæ Mirna Mihovilovic was born on July 13th, 1980 in Split, Croatia. She received her B.Sc. in theoretical physics from University of Zagreb, Croatia. In Fall 2007 she joined the Department of Physics at Brown University as a graduate student. Publications Fabrication of Nanopores with Embedded Annular Electrodes and Transverse CNT Electrodes, Zhijun Jiang, Mirna Mihovilovic, Jason Chan and Derek Stein, Journal of Physics: Condensed Matter, 22, 454114 ( 2010). Statistics of DNA Capture by a Solid-State Nanopore, Mirna Mihovilovic, 2013). Nicholas Hagerty and Derek Stein, Phys. Rev. Lett. 110, 028102 ( Entropic Cages for Trapping DNA Near a Nanopore, Xu Liu, Mirna Mihovilovic and Derek Stein, under review, 2014. Characterizing Non-Equilibrium DNA Dynamics via Delayed Capture and Recapture Using a Solid-State Nanopore, Mirna Mihovilovic, William Poole, Erin Teich and Derek Stein, in preparation. A Nanopore Mass Spectrometer, Joseph Bush, Cole Pruitt, Wooyoung Moon, Mirna Mihovilovic, William Maulbetch, Layne Frechette, Peter Weber and Derek Stein, in preparation. Book Chapters Passive and Electrically Actuated Solid-State Nanopores for Sensing and Manipu- lating DNA, Zhijun Jiang, Mirna Mihovilovic, Erin Teich and Derek Stein in Nanopore-based technology: Single molecule characterization and DNA sequenc- ing, M.E. Gracheva, Ed., Humana Press, Springer, New York ( 2012). iv Talks Non-Equilibrium DNA Dynamics Probed by Delayed Capture and Recapture by a Solid-State Nanopore, Mirna Mihovilovic, Erin Teich, Nicholas Hagerty and Derek Stein, Bulletin of the American Physical Society 57, Number 1, BAPS.2012. MAR.B44.3 (2012); American Physical Society March meeting (Boston, Massachusetts), February 27 - March 2, 2012. The Statistics of DNA Capture and Recapture by Solid-State Nanopores, Mirna Mihovilovic, Erin Teich, Nicholas Hagerty, Jason Chan and Derek Stein, Bul- letin of the American Physical Society 56, Number 1, BAPS.2011.MAR.V43.1 (2011); American Physical Society March meeting (Dallas, Texas), March 21-25, 2011. The Statistics of DNA Capture by a Solid-State Nanopore, Mirna Mihovilovic, Nicholas Hagerty and Derek Stein; Sound bite presentation at the 44th New England Complex Fluids meeting, Brandeis University, September 10, 2010. The Statistics of a Single DNA Capture by a Solid-State Nanopore, Mirna Mi- hovilovic, Nicholas Hagerty and Derek Stein, Bulletin of the American Physical Society 55, Number 2, BAPS.-2010.MAR.P30.1 (2010); American Physical Soci- ety March meeting (Portland, Oregon), March 15-19, 2010. Poster Presentations Studies of Biopolymer Translocations in Solid-State Nanopores, Mirna Mi- hovilovic, Xu Liu, Angus McMullen, Karri DiPetrillo, Lucas Eggers and Derek Stein; National Human Genome Research Institute grantee meeting (San Diego, CA), April 28 - May 2, 2013. Sequencing by Nanopore Mass Spectrometry, Joseph Bush, Mirna Mihovilovic, William Maulbetsch, Carthene R. Bazemore-Walker, Peter M. Weber and Derek Stein; National Human Genome Research Institute grantee meeting (San Diego, CA), April 9-13, 2012. The Statistics of DNA Capture and Recapture by a Nanopore, Mirna Mihovilovic, Erin Teich, Nick Hagerty, Zhijun Jiang and Derek Stein; National Human Genome Research Institute grantee meeting (San Diego, CA), April 4-7, 2011. Won a prize at Institute for Molecular and Nanoscale Innovation 5th anniversary celebration poster competition, Brown University, November 9th, 2012. v The Statistics of DNA Capture by Solid-State Nanopore, Mirna Mihovilovic, Nick Hagerty, and Derek Stein; Bulletin of the American Physical Society 55, Number 13, BAPS.2010.NEF.-B1.12 (2010); Meeting of the New England section of American Physical Society (Providence, Rhode Island), October 29-30, 2010. Conference Contributions Nanopore Mass Spectrometry of DNA Bases, Joseph C. Bush, Mirna Mihovilovic, William Maulbetsch, Layne Frechette, Wooyoung Moon, Cole Pruit, Carthene R. Bazemore-Walker, Peter M. Weber and Derek Stein; Poster at the National Human Genome Research Institute grantee meeting (San Diego, CA), April 28 - May 2, 2013. fd Virus as a Model Sti Polymer for Translocation Experiments with Solid-State Nanopores, Angus McMullen, Xu Liu, Mirna Mihovilovic, Derek Stein and Jay Tang, Bulletin of the American Physical Society 57, Number 1, BAPS.2012. MAR.J50.4, 2012. Teaching Experience Instructor for the course The Magical Inventions of Nikola Tesla, Brown Univer- sity, Summer 2010 and 2013: Conceived and taught a 1-week summer course; through Tesla's inventions students were introduced to basic concepts of electro- magnetism; lead a class of ∼ 15 high-school students as a part of Pre-College curriculum, Oce of Continuing Education. Instructor for the course The Tiniest Bits of Reality, Brown University, Summer 2009 and 2011: Conceived and taught a 2-week summer course on particle physics to a class of ∼ 18 high-school students as a part of Pre-College curriculum, Oce of Continuing Education. Leading Science Club at a community prep school, Spring 2009: Introduced middle- school students to basic physics concepts via paper-cup-scissors experiments, as a part of an ongoing Outreach program at the Brown university's Physics Department, held one-hour weekly meetings. Teaching Assistant for Introductory physics labs, PHYS 60, Spring 2008. Teaching Assistant for Introductory physics labs, PHYS 50/70, Fall 2007. vi Acknowledgements This thesis is a milestone in a long and exciting voyage to which many people con- tributed with their knowledge, skills and support. Along the way, I was shaped and molded though interactions with my mentor Derek Stein, colleagues and friends. My gratitude goes to Derek, who avidly directed my training to become a scientist, by not just guiding me through research but also investing time and eorts to teach me the virtues of a captivating presenter, pragmatic thinker and a dedicated teacher. I thank prof. Jim Valles for his interest in my work, his insightful comments and encouragement in my future endeavors. His presence has always made my presenta- tions at Brown a lot more pleasant and enjoyable experience. I have beneted a lot from having prof. See-Chen Ying on my talks and in my committee, as he has always pushed forward my understanding of polymer theories. The success of the experimental techniques and fabrication procedures is a result of creative contributions from students before me and many who I have worked with in the group. I thank Joe Bush, Jason Chan, Karri DiPetrillo, Nick Hagerty, Daniel Kim, Xu Liu, William Maulbetsch, Angus McMullen, William Poole, Erin Teich and Ben Wiener, who also provided encouragement and a wonderful working environment. Receiving knowledge from one student and passing it onto the next is an important part of a good and close group of colleagues. Our Friday happy hours and group trips helped bridge that relationship to friendship. vii I would like to thank Tony McCormick for enthusiastically introducing me to the world of electron microscopy. Michael Jibitsky was always happy to provide training and re-training for use of various vacuum systems. Charlie Vickers and Mike Packer suered through many designs that I had subjected them to and always managed to provide the parts with thou precision and a big smile. Jerry Zani was always excited to share his numerous gadgets and never failed to amaze me with a demonstration of physics principles at work. Dean Hudek taught me how to present demos to students with enthusiasm of a true discoverer. My stay at Brown would not have been complete without the wonderful times I have spent with my friends. Our road trips, conversations, explorations and ac- tivities have been an indispensable part of my life. Saptaparna Bhattacharya was my shopping companion and fellow science enthusiast. Son Le introduced me to the mysterious world of sh sauce marinades. With Florian Sabou I shared many sunny Sunday afternoons in vigorous conversations on various topics, including dogs, pickling, women and trees and not necessarily in that order. With Alex Geringer- Sameth, my passionate skiing buddy, I made progress in ski turns, and Alexandra Junkes makes the best Caipirinhas in the world, as far as I can remember. Malika and Tomislav Milekovic have become my dearest friends with whom I shared many Saturday mornings over tartine avec du olive. Only thorough unrelenting support, unfailing encouragement and persistent guid- ance of my parents, sister and ujo I have been given the means and opportunity to come this far in my schooling. I thank them for all their strength and all the condence they had and continue to have in me. Finally, to Antun, my superhero. viii Contents Vitæ iv Acknowledgements vii 1 Introduction 1 1.1 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Physics of the DNA macromolecule 6 2.1 DNA is a polymer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.1 DNA as a freely-jointed chain . . . . . . . . . . . . . . . . . . 9 2.1.1.1 The ideal chain . . . . . . . . . . . . . . . . . . . . . 9 2.1.1.2 The real chain . . . . . . . . . . . . . . . . . . . . . 12 2.1.2 The bead-spring model of DNA relaxation . . . . . . . . . . . 13 2.2 DNA is a linear polyelectrolyte . . . . . . . . . . . . . . . . . . . . . 16 2.3 Dynamics of DNA translocation . . . . . . . . . . . . . . . . . . . . . 18 3 Methods for Nanopore Experiments 22 3.1 Fabricating thin membranes on silicon chips . . . . . . . . . . . . . . 23 3.1.1 Fabricating square freestanding membranes . . . . . . . . . . . 25 3.1.2 Fabricating recessed mini-membranes for nanopore drilling . . 26 3.2 Drilling nanopores using a TEM . . . . . . . . . . . . . . . . . . . . . 27 3.3 Preparations for DNA translocation measurements . . . . . . . . . . . 30 3.3.1 Cleaning nanopores . . . . . . . . . . . . . . . . . . . . . . . . 30 3.3.2 Preparing Ag/AgCl electrodes . . . . . . . . . . . . . . . . . . 31 3.3.3 Filtering and degassing buer solutions . . . . . . . . . . . . . 31 3.3.4 Preparing DNA suspensions . . . . . . . . . . . . . . . . . . . 31 3.3.5 Assembling the nanopore chip holder . . . . . . . . . . . . . . 32 3.4 Recording DNA translocations through a solid-state nanopore . . . . 35 3.4.1 Interpretation of measured current through the nanopore . . . 36 4 The Statistics of DNA Capture by a Nanopore 39 4.1 Folded DNA translocations oer clues of DNA capture by a nanopore 40 4.2 Brief description of the experimental setup . . . . . . . . . . . . . . . 42 ix 4.3 Current blockage signals provide snapshots of DNA molecules at the moment of insertion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.4 Measured ECD distributions of folded and unfolded DNA molecules . 44 4.5 Translocation dynamics of folded DNA molecules . . . . . . . . . . . 44 4.5.1 Measured distributions of translocation times . . . . . . . . . 45 4.5.2 Modeling uctuations in the translocation speed . . . . . . . . 46 4.5.3 Possible length-dependence on translocation dynamics . . . . 48 4.6 The statistics of DNA capture by a nanopore . . . . . . . . . . . . . . 49 4.6.1 The measured distribution of capture locations . . . . . . . . . 49 4.6.2 Modeling the capture location distribution . . . . . . . . . . . 51 4.6.3 On the importance of excluded-volume statistics . . . . . . . . 54 4.6.4 Inuence of a length-dependent translocation speed on the dis- tribution of capture locations . . . . . . . . . . . . . . . . . . 56 4.6.5 Inuence of translocation speed uctuations on the distribution of capture locations . . . . . . . . . . . . . . . . . . . . . . . . 56 4.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 5 Non-Equilibrium Dynamics of λ DNA Probed by the Molecular Ping-Pong Technique 61 5.1 Nanopores can probe non-equilibrium congurations of DNA coils . . 63 5.2 A short description of the experimental setup . . . . . . . . . . . . . 65 5.3 Current trace of a ping-pong sequence . . . . . . . . . . . . . . . . . 67 5.4 A brief description of data analysis . . . . . . . . . . . . . . . . . . . 68 5.5 Experimental clues of DNA relaxation from a compressed state . . . . 68 5.5.1 ECD distributions of the rst capture of DNA by the nanopore 69 5.5.2 Measured distributions of recapture times . . . . . . . . . . . 70 5.5.2.1 Modeling recapture time distributions . . . . . . . . 70 5.5.2.2 Comparing predictions of the drift-diusion model to experimental data . . . . . . . . . . . . . . . . . . . 72 5.5.2.3 Recapture time distributions suggest expansion of DNA from an initially compressed state . . . . . . . . . . . 75 5.5.3 Frequent recaptures of molecules with multiple folds suggest more compressed DNA congurations . . . . . . . . . . . . . . 75 5.5.4 Characterizing DNA relaxation with the measurements of av- erage translocation speed . . . . . . . . . . . . . . . . . . . . . 78 5.5.4.1 Modeling the dependence of ECD measurements on the size of DNA coil at the moment of capture . . . . 79 5.5.4.2 The model of DNA relaxation . . . . . . . . . . . . . 83 5.5.4.3 ECD measurements reveal the relaxation time of DNA 84 5.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 x Appendices 87 5.A Description of experimental limitations of the ping-pong technique . . 87 5.A.1 Distributions of measured delay times . . . . . . . . . . . . . . 89 5.A.2 Recaptures within tR -resolution . . . . . . . . . . . . . . . . . 89 5.A.3 Double-trouble . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.A.4 Molecules stuck to the nanopore during translocation . . . . . 93 5.A.5 Distributions of DNA recaptures . . . . . . . . . . . . . . . . 94 5.B Analysis of translocation signals . . . . . . . . . . . . . . . . . . . . . 97 5.B.1 Detection of ping-pong sequences . . . . . . . . . . . . . . . . 99 5.B.2 Fitting the slope of I after the voltage reversal . . . . . . . . . 99 5.B.3 Translocation event detection . . . . . . . . . . . . . . . . . . 102 5.C Numerical evaluation of the drift-diusion equation . . . . . . . . . . 103 6 Outlook 107 Bibliography 112 xi List of Figures 2.1.1 The structure of DNA . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.1.1 Fabricating freestanding silicon nitride membranes. . . . . . . . . . . 24 3.2.1 Milling nanopores through solid-state materials using a TEM. . . . . 28 3.3.1 Assembling the nanopore chip holder. . . . . . . . . . . . . . . . . . . 34 3.4.1 Characterizing DNA by its translocation through a nanopore. . . . . 35 4.1.1 Illustration of DNA captured by the nanopore and ionic current traces from translocation events. . . . . . . . . . . . . . . . . . . . . . . . . 41 4.3.1 Histogram of the number of 20 µs-long current samples. . . . . . . . . 43 4.4.1 Overlaid ECD distributions for translocations of type 1, 2-1, and 2. . 45 4.5.1 Distributions of ttot for events binned by t2 . . . . . . . . . . . . . . . 46 4.5.2 Dependence of httot i on ht2 i. . . . . . . . . . . . . . . . . . . . . . . . 47 4.6.1 Distribution of capture locations. . . . . . . . . . . . . . . . . . . . . 50 4.6.2 DNA capture modeled as polymer tethered to the surface. . . . . . . 52 4.6.3 Distribution of capture locations for the ideal chain. . . . . . . . . . . 55 4.6.4 Distribution of capture locations assuming a length-dependent v. . . . 57 4.6.5 Distribution of capture locations for a uniform distribution of t2 . . . . 58 5.2.1 Illustration of the molecular ping-pong and ionic current trace of 5 successive translocations. . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.5.1 ECD1 histograms of captured molecules. . . . . . . . . . . . . . . . . 70 5.5.2 Histograms of tR measurements for tD between 5 − 150 ms. . . . . . . 71 5.5.3 Numerical solutions of the drift-diusion model. . . . . . . . . . . . . 73 5.5.4 Relative dierence between the mean recapture rates obtained from the drift-diusion model and tR histogram means. . . . . . . . . . . . 74 5.5.5 Illustration comparing the motion of DNA and structureless particle away from the nanopore. . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.5.6 Current blockade histograms and average folding of recaptured molecules. 77 5.5.7 Overlaid histograms of ECDN>1 with histograms of their ECD1 . . . . 80 5.5.8 ECDN>1 /ECD1 histograms. . . . . . . . . . . . . . . . . . . . . . . . 81 5.5.9 ECDN>1 /ECD1 measurements of recaptured molecules as a function of tD + tR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.A.1Current versus time trace of ve consecutive translocations. . . . . . 88 xii 5.A.2Histograms of measured delay times for tD 5−150 ms. . . . . between 90 5.A.3Current trace of a translocation event occurring within 0.2 ms after the voltage reversal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.A.4Current trace of a ping-pong sequence showing capture of two molecules. 92 5.A.5Current trace a ping-pong sequence showing one unusually long translo- cation signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.A.6Histograms show the number of times each molecule got recaptured. . 94 5.A.7Probability of recapturing molecules for the maximum of 26 times. . . 95 5.A.8Probability of recapturing intact λ DNA. . . . . . . . . . . . . . . . . 96 5.B.1Time recoding of 45-second-long current traces. . . . . . . . . . . . . 98 5.B.2Assigning translocations to a ping-pong sequence. . . . . . . . . . . . 100 5.B.3Fitting the slope of I after the voltage reversal. . . . . . . . . . . . . 101 5.B.4Visual output of analysis software. . . . . . . . . . . . . . . . . . . . 102 xiii Chapter 1 Introduction 1 CHAPTER 1. INTRODUCTION A nanopore is a tiny hole in a thin insulating membrane that separates two reser- voirs of saline solution. When a voltage dierence applied between the two reservoirs drives a current of ions through the nanopore, we have a nanometer-sized detector capable of testing physics at scales unobservable to the human eye. Many biologi- cal processes rely on proper functioning and placement of nanopores: cell membrane pores regulate passage of RNA from the nucleus and viruses insert them into the membranes to inltrate the cell with their genetic information. Although life has been thriving o of shuing energy and matter through nanopores for quite a num- ber of years, only recently the means and methods have matured enough to reliably produce nanopores as a robust platform for nanoscale technology. The rst nanopore detectors were made available by embedding an isolated α hemolysin nanopore in an articial lipid bilayer [1], which with its 1.4 nm width is suitable to thread through a single-stranded DNA. As the DNA is negatively charged along its backbone, it is driven through the nanopore by the same electric elds that drive the ionic current. Pulling the polymer through the nanopore, in a process also called translocation, results in disruptions in ionic current. The duration of current disruptions can be used to nd out the length of the molecule that went through. Back in the 1990's this was a revolutionary idea  being able to manipulate sin- gle molecules as they are freely suspended in a solution opened a whole new eld of nanouidics. Since then, the eld has rapidly diversied, leading to solid-state nanopores drilled in tens of nanometers thick silicon nitride or silicon dioxide mem- branes [2, 3], and recently even to nanopores embedded in a single-atom-thin graphene layer [4]. Nanopores have emerged as a truly versatile tool to which electronic, optical and chemical probes can easily be integrated [5]. In these arrangements, they have oered insight into forces acting on polymers at these small scales [6, 7], into drift 2 CHAPTER 1. INTRODUCTION and diusion governing the transport of the DNA towards the nanopore [8], and by analyzing the amplitudes of current disruptions, into the diversity of conformations in which a long DNA threads through the nanopore [9, 10]. These results, along with many others, have energized research towards building an ecient DNA sequencer based on nanopore technology, which relied on the ability to detect properties of DNA molecules while inside the nanopore. Surprisingly enough, our experiments have shown that current blockade signatures produced by the DNA's presence inside the nanopore can also deliver information on what the molecule was doing before being captured by the nanopore. By studying the time it takes the DNA to arrive at the nanopore, by counting the number of times it folded while entering and by measuring the speed at which the DNA translocates through the nanopore, we can access a number of fundamental polymer studies. For example, we will show that the conformations at which the polymer presents itself to the nanopore entail knowledge of the physics that is relevant at these scales. Further- more, we can infer the state in which the DNA emerges from the nanopore, as well as probe the dynamics of long polymer molecules in solutions. These observations lie at the heart of understanding nanouidic phenomena and can oer clues to the physics of life at its smallest scales. 1.1 Outline of the thesis Threading the DNA through nanopores is sensitive to polymer physics that takes over at nanometer scales. In Chapter 2, we will therefore introduce statistical prop- erties of DNA as a polymer. Most of the theories describing physical properties of polymers were already familiar, but managed to evade experimental verication at 3 CHAPTER 1. INTRODUCTION the level of single molecules for more than 30 years. We will also briey introduce the electrostatics of charged DNA immersed in an electrolyte and the forces acting on the DNA as it traverses the nanopore. Early experiments found a surprising su- perlinear dependence of measured translocation times on the DNA length [9, 10, 11], which motivated a plethora of theories that seek to model the translocation process [12, 13, 14, 15, 16, 17], some of which are reviewed at the end of this chapter. We will nd that our experiments can provide more insight to the dynamics of translocations. In Chapter 3 we describe the materials and methods that we developed to produce solid-state nanopore detectors. We describe detailed procedures, starting from fab- rication of membranes, drilling of the nanopores, and the experimental preparations for the assembly of a nanopore detector. These methods were also published as a book chapter in [18]. By applying the voltage dierence across the nanopore device we detect individual DNA molecules. We conclude the chapter with description of translocation signals. In Chapter 4 we characterize the capture of the DNA by the nanopore. As the DNA inserts itself somewhere along its length into the nanopore, we nd a strong bias that the insertion happens close to the DNA's end. We published these ndings in [19]. This simple observation is shown to be a consequence of the equilibrium distribution of polymer congurations outside the nanopore. By modeling the DNA capture as polymer tethered to a surface at a single point, we show that the cong- urational entropy is sucient to explain the observed preponderance of end capture. In this work we also study the inuence of velocity uctuations on the translocation dynamics, showing that we can utilize nanopores as a microscope precise enough to map lengths along the DNA. 4 CHAPTER 1. INTRODUCTION In Chapter 5 we utilize a novel technique, called molecular ping-pong [8], to learn more about the dynamics of DNA coils outside the nanopore. We nd that the translocation speed of identical DNA molecules reects the size of the DNA coil at the onset of translocation, mapping our translocation signals to relaxation dynamics of the DNA polymer. We infer that the two major contributions to the drag force, the drag on the segment inside the nanopore and on the segments moving towards it, are sucient to explain the trends we observe in the translocation speeds of molecules recaptured at various times after translocation. Each recapture of the DNA measures its state of relaxation and places it again in an out-of-equilibrium compressed state. By utilizing nanopores as a pump-probe device, our measurements paint a simple picture of a DNA relaxing from an initially compressed conguration induced by translocation through the nanopore. We nd a simple model of coil relaxation explains the data remarkably well. These results are being prepared for publication [20]. Finally, in Chapter 6 we conclude by outlining some future directions and possible applications of the techniques presented herein. 5 Chapter 2 Physics of the DNA macromolecule 6 CHAPTER 2. PHYSICS OF THE DNA MACROMOLECULE Our nanopore detectors, while threading a long DNA molecule through, inte- grate the signals of around 60 base-pairs that are at any moment present inside the nanopore. For a many-thousand base-pairs long DNA, our experiments have shown that the polymer properties of the DNA outside the nanopore also inuence the translocation signals. Here we review some known physical models of polymer chains in solutions. Since the DNA is charged in an ionic solution and in our experiments moves under the inuence of an external electric eld, in this chapter we also briey introduce electrostatic properties of polyelectrolytes. We end the chapter with a review of known models of translocation dynamics. 2.1 DNA is a polymer The DNA molecule has two strands wrapped around each other, forming a 2.2 nm- wide helix. As shown in Fig. 2.1.1, each strand consists of repeating units, called nucleotides, which are made up of a charged phosphate group, a deoxyribose sugar and one of the 4 possible bases: Adenine, Guanine, Cytosine or Thymine. The bases on opposite strands are connected into a base-pair through hydrogen bonds. The phosphate-deoxyribose backbone carries one electron per each phosphate group, which amounts to 2 electrons per base-pair, with average distance between base-pairs of 0.34 nm. Rather than probing the detailed chemical structure of DNA, our experiments treat DNA mechanically, allowing us to think of it as a sequence of structureless units, monomers. As we walk along its contour length, we will nd that the initial orientation of monomers persists for 50 nm, or around 150 basepairs. This distance along the chain is called the persistence length, lP , and over this length correlations 7 CHAPTER 2. PHYSICS OF THE DNA MACROMOLECULE Figure 2.1.1: The structure of DNA. The charged phosphate-deoxyribose backbone carries covalently linked bases, with two strands connected through hydrogen bonds across Adenine-Thymine and Cytosine-Guanine base-pairs. Credit: Madeleine Price Ball, http://en.wikipedia.org/wiki/DNA 8 CHAPTER 2. PHYSICS OF THE DNA MACROMOLECULE between the rst and last monomer decrease by 1/e. The persistence length can be related to the bending rigidity κB of the DNA via lP = κB /kB T , where kB is the Boltzmann constant, and T temperature; it is the characteristic length scale at which thermal uctuations take over, causing the DNA to bend and coil. Since the 16.5 µm long double-stranded λ DNA, phage of virus we use in our experiments, is much longer than lP , it forms a random coil in a solution. In our experiments we use nanopores to probe the statistical and dynamical properties of DNA coils, which we will introduce in the next section. 2.1.1 DNA as a freely-jointed chain More than thirty years ago, a novel way of treating polymer physics emerged, drawing its strengths from phase transitions in condensed matter systems, giving rise to simple scaling relations and estimating critical exponents. The critical exponents describe universal features of a system, and in the case of a polymer chain, such as DNA, we can restrict to only two, γ and ν. The exponent γ is related to the chain's entropy, and ν to its size. In the following we will introduce the basic concepts behind the physics governed by these two exponents, adopting exposition found in [21]. 2.1.1.1 The ideal chain In the simplest picture the DNA can be thought of as a chain of N segments of length lK described as a random walk on a periodic lattice. On the lattice, the DNA takes a step in any direction towards a neighboring site with equal probability, where each step is completely uncorrelated with the previous one. The length of the step corresponds to the length along the molecule for which all correlation between monomers is lost. It equals twice the persistence length and is called Kuhn length, 9 CHAPTER 2. PHYSICS OF THE DNA MACROMOLECULE which for the double-stranded DNA is lK = 100 nm. After N = 165 steps of length lK , the λ DNA forms a random coil. In this context the DNA is referred to as an ideal or Gaussian chain. This approach allows us to access many statistical properties of the chain through analytical and numerical computations of random walks. The size of an ideal coil Starting from a xed point on the lattice, after N steps the DNA ends its random walk at position ~r, which is a sum of steps of length lK , ~r = ~l1 + ~l2 + . . . + ~lN . The average size of the DNA coil is then given in terms of the average square end-to-end distance of N random walks. Since all steps are independent, we get X X R02 = h~r 2 i = h~ln~lm i = h~ln2 i = N lK 2 (2.1.1) n,m n Therefore, the average size of the ideal chain scales as the square root of the number of steps, R0 = lK N 1/2 . Another characteristic length scale important for polymers in solution is the radius of gyration, Rg , which is the second moment of mass distribution, and can be calculated as a sum over mean square distances between all segments of the chain, 1 X ~ ~ 2 Rg2 = ln − lm (2.1.2) 2N 2 n,m √ In equilibrium it is related to the average end-to-end distance with Rg = R0 / 6. We will frequently use Rg as a measure of size of DNA in solution. For λ DNA in equilibrium, Rg ' 600 nm [7]. Number of microstates The number of conformations the DNA can take starting from a xed point on the lattice and ending at ~r is given by the number of distinct 10 CHAPTER 2. PHYSICS OF THE DNA MACROMOLECULE walks, ΩN (~r). The total number of conformations of a polymer will simply be X ΩN = ΩN (~r) = µN (2.1.3) ~ r since in each step there are µ independent possible orientations. For example, for the cubic lattice, µ = 6, and the total number of microstates for λ DNA is 6165 ' 10128 . The elasticity of the ideal coil For a large number of steps, N  1, the proba- bility distribution of ~r is a Gaussian: 3r2 3r2 p(~r) ∝ N −3/2 exp(− ) = N −3/2 exp(− ) (2.1.4) 2h~r 2 i 2R02 where the number 3 in the exponent comes from considering a walk in 3-dimensional space. Since p(~r) = ΩN (~r)/ΩN and the entropy is S(~r) = kB ln ΩN (~r), we get that the entropy decreases as the polymer stretches: 3r2 ∆S = S(~r) − S(0) = −kB (2.1.5) 2R02 The free energy, F = E − T S, associated with deformations of the initial size of the polymer coil is then 3kB T 2 Felastic (~r) = F (0) + r (2.1.6) 2R02 The rst derivative of the free energy is the force, in this case the hookean spring force fspring = −κr, from which we can extract the spring constant of an ideal coil, κ = 3kB T /R02 . 11 CHAPTER 2. PHYSICS OF THE DNA MACROMOLECULE 2.1.1.2 The real chain Experiments that measured size of the polymer coils in dilute solutions found that their radius of gyration is slightly larger than expected for an ideal chain [21]. The swelling of the coil is explained by excluded volume interactions, which in turn need to enter in the random walk model. The DNA can then be modeled with a self- avoiding-walk (SAW) on a periodic lattice, which forbids the polymer from taking a step where the lattice site is already occupied by another segment. These models can no longer be analytically studied, so numerical methods have been developed to access statistical properties of these systems. Size of the real chain The average end-to-end distance of a real chain is RF = lK N ν , where the dierence from the ideal chain is encoded in the exponent ν. As the chain cannot fold back on itself, in the same number of steps it will on average cover a larger area than the ideal chain, so ν > 1/2. The subscript F is to denote this scaling is due to Flory, whose simple argument is described below. Suppose a spherical coil of radius R has monomer concentration c = N/R3 . Due to monomer interactions, there is repulsive energy in the chain, proportional to the number of interacting pairs per unit volume, F = kB T vc2 /2. Here, v is the excluded volume parameter, and c2 counts the number of pairs within unit volume of coil. If all correlations between monomers are ignored, we can replace hc2 i with hci2 , where the average is taken within volume of the coil. Integrating F over the volume of the coil, we have N2 Frepulsive ∝ kB T vc2 R3 = kB T v (2.1.7) R3 By summing Frepulsive , which tends to swell the chain, with Felastic (Eq.2.1.6) we 12 CHAPTER 2. PHYSICS OF THE DNA MACROMOLECULE obtain Ftot N2 R2 'v 3 + 2 (2.1.8) kB T R lK N 2/5 for which there is a minimum at RF = v 1/5 lK N 3/5 , or RF ∼ N ν , where ν = 3/5. Most recent numerical and experimental estimates of ν lie within a few percent of the value obtained by this simple scaling argument. Flory's idea seems to work as two wrongs conspire to make right. First, the repulsive energy is erroneously overestimated, as correlations between segments are omitted, and second, the elastic energy is overestimated, as it is borrowed from the treatment of the ideal chain, Eq. 2.1.6. Number of microstates of the real chain The total number of SAWs of N steps, at N large, asymptotically approaches ˜N N γ−1 ΩN = const. µ (2.1.9) where µ ˜ depends on the choice of lattice, and the case of cubic lattice gives 4.68, a somewhat smaller value than µ = 6. γ = 7/6 is a universal scaling exponent, which depends only on the dimensionality of the system. N γ−1 is called an enhancement factor, and we will say more about these factors when we start describing polymer conformations in solution, in Ch.4. 2.1.2 The bead-spring model of DNA relaxation In our experiments in which we implement the molecular ping-pong technique, we access dynamics of polymer relaxation outside the nanopore and measure the time τ in which the whole coil re-equilibrates after translocation. Here we review the Rouse 13 CHAPTER 2. PHYSICS OF THE DNA MACROMOLECULE model which describes the dynamical properties of polymers that are randomly coiled in solutions [21]. In this model, the polymer consists of N beads connected by hookean springs, with force fspring = −κ∆x where κ = 3kB T /l2 is the spring constant and ∆x is the spring extension from equilibrium length l . The positions of beads are denoted with ~r1 , ~r2 , . . . , ~rN . Each bead experiences a force from its neighbors 3kB T f~n = 2 [~rn+1 + ~rn−1 − 2~rn ] (2.1.10) l where the term in the brackets in the limit of large number of beads can be replaced by ∂ 2~rn /∂n2 . Balancing the elastic spring force with the viscous drag force on the bead, f~n = ζb ∂~rn /∂t, where ζb is the drag coecient of the bead, we get ∂~rn 3kB T ∂ 2~rn = (2.1.11) ∂t ζb l2 ∂n2 For each bead of radius Rb , its drag coecient will be ζ = 6πηRb , where η is solvent ∂~ rn ∂~ rn viscosity. Supplemented with boundary conditions | ∂n n=0 = | ∂n n=N = 0 at the ends of the polymer, Eq.2.1.11 is a one-dimensional wave equation, where the solutions can be expressed in terms of eigenmodes ~rn,p πpn ~rn,p (t) = α ~ p cos exp(−t/τp ) (2.1.12) N where α ~p is the amplitude, and τp is the characteristic relaxation time of the p-th mode, ζb l2 N 2 τp = (2.1.13) 3π 2 kB T p2 14 CHAPTER 2. PHYSICS OF THE DNA MACROMOLECULE Therefore, the longest relaxation time (for p = 1), the time of changes in conformation of the polymer on the order of the size of the coil, scales as τ1 ∝ N 2 . In the Rouse model each bead responds to the motion of its rst neighbors through elastic spring interactions. However, whenever we apply a force to a bead somewhere along the chain, the motion of that bead also creates the uid velocity eld. The surrounding beads are consequently inuenced by that velocity eld and experience hydrodynamic drag. In the Rouse approximation, the chain is assumed to be freely drained by the solvent, that is to say the hydrodynamic interactions are ignored. Furthermore, by taking the spring force of an ideal chain, the Rouse model ignores excluded volume interactions. Once the hydrodynamic interactions are taken into account, the whole random coil behaves as an impermeable sphere of radius R. The drag coecient is then given by ζ = 6πηR. That is to say, the coil carries the uid within it and the drag force on the coil is only coming from the outside. By using the Einstein relation, ζ = kB T /D, where D is the diusion constant, we can obtain the Zimm relaxation time, the characteristic time it takes the coil to diuse its own radius R: R2 R3 τ= = 6πη (2.1.14) D kB T If we substitute for R the size of the swollen chain RF ∝ N ν , this gives τ ∝ N 1.8 . This result is not far from the Rouse τ ∝ N2 scaling. Therefore, to a large extent, the corrections due to hydrodynamic interactions and the excluded volume cancel out. 15 CHAPTER 2. PHYSICS OF THE DNA MACROMOLECULE 2.2 DNA is a linear polyelectrolyte Our nanopore experiments use the fact that the DNA molecule in solution is highly charged. However, this also modies its chain statistics. The short range electro- static repulsion along the DNA contributes to its stiness, therefore increasing the DNA's persistence length, and consequently the coil's size. For example, the DNA in 100 mM solution will have persistence length of 52 nm and at 1 M lp drops to 43 nm [22]. Furthermore, the eective diameter of the polymer decreases with increasing salt concentration, from 5.5 nm at 100 mM to 2.7 nm at 1M [22]. In this way, the electro- static interactions contribute to excluded volume interactions and result in swelling of the chain. These eects are strong at low ionic concentrations, and weak at high ionic concentrations, the latter being the conditions at which we do our experiments. In order to explain why the electrostatic eects are weak at high ionic concentra- tions, we need to address the electrostatic properties of charged polyelectrolytes in ionic solutions. Because the ions are freely roaming around the solution, they will want to screen the excess negative charge on the polymer. To estimate the thickness of the layer of counter-ions, we model the electrostatic potential φ(r) around the DNA with the Poisson equation: ∇2 φ = −ρ/ (2.2.1) where ρ = e(n+ − n− ) is the mobile ion charge density. The volume concentration n± of mobile ions in thermal equilibrium, located within potential φ(r), obeys the Boltzmann statistics: n± = n0 exp(∓eφ(r)/kB T ) (2.2.2) where n0 is the initial volume concentration of co-ions and of counter-ions. Equa- 16 CHAPTER 2. PHYSICS OF THE DNA MACROMOLECULE tion 2.2.1 in combination with Eq. 2.2.2 leads to a non-linear Poisson-Boltzmann equa- tion, en0 ∇2 φ = 2 sinh(eφ/kB T ) (2.2.3)  At low concentrations, that is large distances between ions, we can expand the sinh function in Eq. 2.2.3 to obtain a linearized equation: ∇2 φ = κ2D φ (2.2.4) where κ2D = 2e2 n0 /kB T . In literature, 1/κD is called the Debye screening length. To get a sense for what this means, we can solve Eq.2.2.4 in 1D where we get a typical exponential solution φ(r) ∝ exp(−κD r). Hence, 1/κD measures the range of electrostatic eects in solution due to ions. Beyond 1/κD , the interactions are p eectively screened. In ionic solution at 25 0 C, 1/κD = 0.3 nm/ I(M ), where I(M ) is the ionic strength expressed in molar units. In our experiments at 1 M KCl solution, therefore, the concentration of the mobile ions in the uid around the DNA can be taken as approximately constant, with a very narrow, 0.3 nm-wide Debye layer around the polymer. This means that the DNA charge is eectively screened over very short distances, and the role of DNA-DNA electrostatic interactions is suppressed, even when two DNA segments are relatively close to each other. At lower concentrations the screening is not as eective, and for 0.1 M solution the Debye layer is 1 nm wide, which is on the order of the radius of DNA. Therefore, both the persistence length and the eective diameter of the polymer increase with the thickness of the Debye layer [22]. 17 CHAPTER 2. PHYSICS OF THE DNA MACROMOLECULE 2.3 Dynamics of DNA translocation In nanopore experiments, the applied voltage generates an electric eld across the nanopore that acts only on a short length of the DNA inside or very near the nanopore. Translocation of DNA is a dynamical process where a number of eects are happen- ing simultaneously. As the DNA electrophoretically threads through the nanopore, the counter-ions along its backbone are moving in opposite direction, inducing an electroosmotic ow of the solvent around DNA. This eectively reduces the driving force of the DNA. Furthermore, similarly to DNA, the SiN nanopore walls at pH 8 carry negative charge density of about 30 mC/m2 [23], which is again screened by counter-ions, whose motion in an electric eld additionally acts counter to the elec- trophoretic force on DNA [6]. Simply written, the eective driving force on the DNA inside the nanopore, fE = fES − fEO , has two contributions: one from the electro- static force on the backbone, fES = λ∆V , where λ is the DNA's linear charge density and ∆V potential dierence applied across the nanopore, and the other from fEO , the electroosmotic force arising from the motion of counter-ions, which is also pro- portional to ∆V . Therefore, the eective driving force can simply be expressed as fE = λef f ∆V , where λef f is the reduced DNA linear charge density. Van Dorp et al. have utilized the optical tweezer setup together with nanopores to measure those picoNewton forces at low salt concentrations, and found that the eective driving force on the DNA lodged inside a 10 nm nanopore is ∼ 20 pN at ∆V = 100 mV [6]. The eective driving force is balanced by the viscous drag force, which acting on the straight segment of DNA inside the nanopore can be approximated with drag on the cylinder moving through uid times the velocity of the DNA, fD = ζpore v . In a typical nanopore setup, a 16.5 µm-long λ DNA translocates the nanopore in 2 ms, 18 CHAPTER 2. PHYSICS OF THE DNA MACROMOLECULE which gives v ' 10 mm/s. We can then use the formula supplied by [6] and [24, 25, 26] 2πηLpore fD = v (2.3.1) ln(rpore /rDN A ) where Lpore = 20 nm is the nanopore thickness, η = 10−3 Ns/m2 and rpore /rDN A ' 5. However, typical values lead to fD ' 0.5 − 1 pN, which is an underestimate of an order of magnitude when compared to experimentally measured fE [6], and which may hint that additional drag is needed to balance the electrophoretic force on DNA. If the drag inside the nanopore were the only contribution to the viscous drag force on the DNA, from the force balance fE = fD , we would obtain v = λef f ∆V /ζpore . Since the drag coecient ζpore does not depend on the length of DNA, in experiments we would expect to observe a linear relationship between end-to-end translocation time and length of the molecule, Ttr ∝ L. Surprisingly, the mean translocation time has been shown to increase superlinearly with length of DNA, as Ttr ∝ Lα with α = 1.27 ± 0.03 [10, 11]. Furthermore, experiments by Lu et al. have shown that the uctuations in Ttr are much larger than what we would expect based on only Brownian velocity uctuations [13]. Both of these eects can be accounted by the drag on the DNA coil outside the nanopore. Additionally, simulations by Lu et al. have showed that longer translocation times of identical DNA molecules may arise from unraveling dynamics of more extended initial conformations of the DNA coil outside the nanopore, while shorter measurements of Ttr could be correlated with DNA initiating translocation from more compressed congurations. These results suggest that nanopores, although they can only detect DNA when it is inside the nanopore, in fact have access to information on what the DNA looks like and what it is doing outside the nanopore. 19 CHAPTER 2. PHYSICS OF THE DNA MACROMOLECULE Storm et al. explained the exponent α to come from the scaling Ttr ∝ L2ν = L1.2 , where the polymer chain is treated as a macroscopic blob, whose size is quasi- statically reducing as the polymer gets gradually transferred through the nanopore [11]. However, an over-simplied picture like this is most likely to be incorrect. The drag on the segments outside the nanopore acts on distances along which the segment is being pulled into the nanopore, not on the polymer coil on the whole. Storm et al. results have motivated a wealth of translocation models, seeking to predict the value of α from rst principles. Grosberg et al. modeled the pulling of the polymer into the nanopore as a sequen- tial process of straightening the folds of the initial polymer conguration [12]. As the DNA is pulled into the nanopore, the moving end of the randomly coiled strand starts unfolding, and only the folds closest to it start unraveling, while the remainder of the coil stays locked in place. This process is analogous to dropping a length of rope from the side of the table. They found scaling Ttr ∝ L1+ν when excluded volume eects are taken into consideration. A somewhat more robust description is supplied by Sakaue et al., in which the polymer outside the nanopore has a moving part and a dormant part, with the bound- ary, B(t), between the two moving in time as the translocation is in eect [15, 16]. The moving part is the part of the polymer that is extended and in the force driven regime forms a so called trumpet, while the dormant part corresponds to part of the poly- mer unperturbed by the translocation process. Matching the boundary conditions at B(t) then leads to the estimate of the translocation time by solving B(Ttr ) = Rg , and to the estimate of α = 1 + ν. Other models have also been proposed, in which the DNA funnels through a xed trumpet shape, where they assume that the ux of monomers is constant along any 20 CHAPTER 2. PHYSICS OF THE DNA MACROMOLECULE section of the funnel [17]. Other congurations are also possible, like the stem-ower conguration at high electric eld strengths [27], where the monomers in the ower are fed to the nanopore along the stem. The jury is still out on the precise understanding of the dynamics of translocation, providing still a rich and complex playground for many theories, numerical simulations and phenomenological models. In this thesis we will describe experiments that can elucidate some of the statistical and dynamical properties of a large λ DNA coil, by sliding it through a nanopore. 21 Chapter 3 Methods for Nanopore Experiments 22 CHAPTER 3. METHODS FOR NANOPORE EXPERIMENTS In this chapter, we describe the materials and methods we employ in creating and using solid-state nanopores. Our fabrication methods are largely based on those developed by Krapf et al. [28], which we also described in [18]. The rst steps, illustrated in Fig. 3.1.1, use standard microfabrication techniques [29, 30] to create 20 nm-thin freestanding silicon nitride membrane. Single-nanometer-scale pores are created by drilling through the thin membrane using a transmission electron micro- scope (TEM) with a eld-emission source and an acceleration voltage of at least 200 kV [Fig. 3.2.1] [2]. We also describe our methods for using solid-state nanopores to detect DNA translocations. These measurements require signicant preparations to clean the nanopore chips and holders, degas the solutions, and prepare the electrodes, only after which can the nanopore be assembled in its holder, as shown in Fig. 3.3.1. Then we describe the procedures for introducing DNA and measuring translocations. 3.1 Fabricating thin membranes on silicon chips The starting point for our solid-state nanopore devices is a 400 µm-thick, double-side polished, silicon (100) wafer coated with thin lm materials. We call the two sides of the wafer the at and the pit. The at side refers to the at freestanding membrane, and the pit to the opening in the silicon that leads to the freestanding membrane. The material stack on the at side of the silicon wafer begins with 20 nm of low-stress silicon nitride (SiN) grown by low-pressure chemical vapor deposition (LPCVD) on silicon, followed by 400 nm of silicon dioxide (SiO2 ) grown by plasma enhanced chemical vapor deposition (PECVD), and nally 400 nm of SiN grown by LPCVD. The pit side of the wafer is coated only with 420 nm SiN by LPCVD. 23 CHAPTER 3. METHODS FOR NANOPORE EXPERIMENTS Figure 3.1.1: Fabricating freestanding silicon nitride membranes. (a) The proce- dure is illustrated by cross-sectional images showing the silicon chip: i) following photolithography and reactive ion etching through silicon nitride, which transfers a 610 × 610 µm square to the pit side; ii) after KOH etching creates the pit, leaving a 50 µm-wide freestanding window on the at side; iii) after a 5 µm-diameter circle is patterned on the at side by photolithography and etched through to the SiO2 layer by RIE; and iv) after BHF selectively removes the exposed SiO2 , leaving a 20 nm- thick freestanding circular mini-membrane. (b-e) Optical microscope images acquired during a successful and (f-i) unsuccessful fabrication run: (b) the 610×610 µm square transferred to the pit side ((f ) disrupted RIE etching), (c) the silicon pit leading to a freestanding window following the KOH etch ((g) rough side walls, possibly due to temperature variations), (d) the intact mini-membrane viewed from the at side ((h) broken mini-membrane due RIE or BHF overetching), and (e) the freestanding window viewed from the at side ((i) white rim is due to lateral etching of SiO2 by 40% HF etch). 24 CHAPTER 3. METHODS FOR NANOPORE EXPERIMENTS 3.1.1 Fabricating square freestanding membranes We noticed that good electrical conductance of our nanopores heavily relies on the cleanness of the surface. We therefore start the rst steps of our fabrication process by carefully cleaning the wafer by sonicating it in acetone, then isopropanol, then DI water. During the fabrication it is important to rinse the wafer under running deionized water after each step, in order to ush away the impurities. Via photolithography, we transfer a pattern of 6-by-6 array of 610 µm squares spaced 5 mm apart, together with a rectangle grid dening the boundaries of each 5 × 5 mm nanopore chip. This patterned surface is to become the pit side. After successful photolithography the pattern of squares becomes clearly visible on the wafer. We use an optical microscope to check that the edges of the patterned squares are sharp. We then etch through 420 nm-thick SiN layer using a reactive ion etcher (RIE) with CF4 process gas, as illustrated in Fig. 3.1.1 (a) i). After the etch, the patterned squares appear silver in color, indicating that the etch went through the SiN to the underlying Si, as shown in Fig. 3.1.1 (b). We conrm the depth of the etched pattern with a stylus prolometer, and from the measured depth and the etch time we calibrate the etching rate, which we will use again later. The longest step of our fabrication involves the anisotropic wet KOH etch which removes 400 µm of Si and exposes the pit side of the 20 nm-thin SiN membrane, illustrated in Fig. 3.1.1 (a) ii). The etching rate of Si in 40% KOH solution depends strongly on the solution temperature (at 85◦ C the rate is 87 µm/hour and at 70◦ C it is 35 µm/hour), so it is important to precisely control the temperature during the etch. In addition to using a hot plate with the feedback temperature sensor, we found that placing a Teon beaker lled with KOH solution in an oil bath (another large 25 CHAPTER 3. METHODS FOR NANOPORE EXPERIMENTS beaker lled with oil) does the trick. After ∼ 13 hours at 70◦ C we etch through the entire Si substrate. Since the freestanding membranes are made of transparent materials, completion of the KOH etching process can be seen by the appearance of a square array of small transparent spots on the at side of the wafer. Due to anisotropic etch, the lateral size of each transparent membrane at the bottom of the pit is approximately 50 µm, which we conrm using an optical microscope. The membrane should be square, and the walls of the Si pit leading to it should appear clear and smooth, as shown in Fig. 3.1.1 (c). 3.1.2 Fabricating recessed mini-membranes for nanopore drilling We continue the fabrication on the other, at, side of the wafer by rinsing rather than sonicating the wafer in acetone, isopropanol, and then DI water; sonication would break the 800 nm-thin membranes. Via photolitography, we pattern the 6-by-6 array of 5 µm-diameter circles on the at side of the wafer by aligning each circle on the mask with the center of a square membrane below. We again use RIE to etch approximately 600 nm into the at side, based on the previously calibrated etching rate, as illustrated in Fig. 3.1.1 (a) iii). If the membrane gets completely etched through, it will appear as a black circle in the middle of the square window as shown in Fig. 3.1.1 (h). We selectively etch away the remaining SiO2 layer, using buered HF (BHF), to isolate a 20 nm-thin SiN mini-membrane, depicted in Fig. 3.1.1 (a) iv). Optical tomograph in Fig. 3.1.1 (d) shows a light-blue 20 nm-thin and 5 µm-wide SiN mini- membrane, obtained by dipping our wafer in BHF for no more than 4 minutes. We found out the hard way that BHF also slowly etches SiN at a rate of less than ∼ 0.5 nm/min at room temperature. If we keep the wafer in BHF for too long, it will 26 CHAPTER 3. METHODS FOR NANOPORE EXPERIMENTS completely etch through the thin SiN membrane and the result will be the same as in Fig. 3.1.1 (h). After etching through the SiN membrane, it will also start laterally etching the SiO layer, creating a white rim around the square membrane window as in Fig. 3.1.1 (i). In our lab, this ended up being a useful technique to fabricate structures on top of nanopores. We then divide the wafer into individual nanopore chips by cleaving it along the horizontal and vertical trenches. Before drilling a nanopore in a TEM, we clean the chips with Nanostrip to remove impurities that might have been left over from the fabrication process or handling the wafer. This nal cleaning step in our fabrication procedure has been shown to improve yield of low noise nanopores. 3.2 Drilling nanopores using a TEM Figure 3.2.1 (a) illustrates the focused beam of electrons from a TEM piercing a nanopore through a 20 nm thin mini-membrane. The drilling process relies on sput- tering, chemical deposition and thermal evaporation to displace atoms when the fo- cused beam of electrons impinges on the membrane [2]. We use a JEOL 2010F with a eld-emission source and an acceleration voltage of at least 200 kV. We insert the nanopore chip in TEM via the custom made brass TEM holder, shown in Fig. 3.2.1 (b). If the fabrication was successful and the mini-membrane is thin enough, it will appear as a bright dot on the phosphorus screen, even at the lowest 150× magnication. As we increase the magnication to 500 000×, we center the mini-membrane for as long as its perimeter is entirely visible. In this way we make sure to drill the nanopore in the center of 5 µm mini-membrane, which greatly helps if we want to nd the nanopore to image it after the experiment. 27 CHAPTER 3. METHODS FOR NANOPORE EXPERIMENTS Figure 3.2.1: Milling nanopores through solid-state materials using a TEM. (a) Schematic of the focused beam of electrons from a TEM piercing a nanopore through a thin membrane. (b) A brass sample mount used to introduce 5 × 5 mm nanopore chips into the TEM. (c) TEM micrographs showing underfocused (top), overfocused (middle) and in-focus (bottom) ∼ 100 nm hole in carbon lm. In the left column are images with corrected objective lens astigmatism; right column shows distorted images due to objective lens astigmatism. (d) TEM micrographs of a 10 nm-diameter nanopores in a SiN membrane made by TEM milling showing a round, symmetric nanopore (left) and elongated nanopore due to sample stage drift (right). Panels (a, b) are from [18]. 28 CHAPTER 3. METHODS FOR NANOPORE EXPERIMENTS We carefully adjust the condenser stigmation to obtain a tight, symmetrical, and bright spot with the beam fully condensed, which at the end determines the shape of the nanopore. It is important to do this adjustment quickly and spread the beam so as not to accidentally drill a nanopore through the membrane. It is also important to correct for objective astigmatism which at the end determines the quality of the image of the nanopore. As an example, in Fig. 3.2.1 (c) we show through focus im- ages (underfocused, overfocused and in-focus images) of 100 nm holes in carbon lm with well adjusted objective stigmation (left column) and poorly adjusted objective stigmation (right column). We then proceed to drill a nanopore by condensing the beam to a narrow spot. As the spot size is reduced, it is still possible to resolve the grainy surface of the membrane, which appears to sparkle when the spot is fully condensed. This will indicate that the beam is well focused during the drilling of the pore. It takes about 30 seconds to drill a ∼ 10 nm diameter pore such as shown in Fig. 3.2.1 (d). Note that sample might drift during these 30 sec, in which case the pore will be elongated [Fig. 3.2.1 (d), right]. This step requires some practice to keep the membrane in focus, to condense the beam to the right extent, and to keep its position steady. We have noticed that we seem to enjoy greater success at measuring DNA translo- cations through nanopores that were drilled and removed from the TEM very quickly, as compared with pores that were exposed to the electron beam for much longer times as the result of a need to adjust the focus or stigmation, or to acquire multiple images. Our evidence is mainly anecdotal, but it has led us to suspect that long exposures to the electron beam contaminate the surface of the mini-membrane and the nanopore with hydrocarbons that degrade their wettability or electrical properties. 29 CHAPTER 3. METHODS FOR NANOPORE EXPERIMENTS 3.3 Preparations for DNA translocation measurements Here we describe preparations and assembly of the nanopore chip into its holder, in a way we found gives the highest yield of electrically quiet nanopores that detect translocations. We estimate that the overall yield of nanopore devices that success- fully register DNA translocations, when prepared this way, is approximately 25%. Common failure modes include: (i) damage to the nanopore chip before it can be tested in an ionic current measurement, (ii) excessively high levels of electrical noise that would obscure DNA translocation signals, (iii) electrically quiet pores that be- come quickly blocked as a DNA molecule presumably inserts and sticks in the pore, and (iv) electrically quiet pores that never register a DNA translocation for reasons that are still unclear to us. It has been our experience that when a nanopore fails for one of these reasons, it is relatively rare that cleaning the chip again and re- assembling the chip holder will revive it. Our practice has been to simultaneously clean and assemble 3 or 4 nanopore chip holders, test each one, and simply discard the nanopores that are uncooperative. Nanopores that register DNA translocations with low noise can usually be cleaned and successfully re-used several (∼ 4) times. 3.3.1 Cleaning nanopores We clean the nanopores with Nanostrip solution, which removes organic contami- nates and makes the nanopore hydrophilic. To activate Nanostrip, we heat it to 80◦ C and clean the nanopore chips, usually 4 at a time, for 2 hours. This cleaning procedure greatly reduces high levels of electrical noise, suggesting that hydrophobic contaminants are at its root. 30 CHAPTER 3. METHODS FOR NANOPORE EXPERIMENTS 3.3.2 Preparing Ag/AgCl electrodes To prepare the electrodes' surface with AgCl, which establishes a well dened reference potential in KCl solution, we rst gently scrape o any existing AgCl layer using a razor and then immerse the tips of Ag electrodes into bleach. The tips of the electrodes develop a dark layer of AgCl. 3.3.3 Filtering and degassing buer solutions We remove contaminants larger than 0.22 µm by ltering the solutions we use in our experiment, which are: DI water, 10 mM Tris - HCl (pH 8), and high-salt 1M KCl (with 10 mM Tris - HCl, 1 mM EDTA, pH 8). Immediately before starting an experiment, we also degas our solutions in a desiccator held under vacuum. We nd that this helps prevent bubbles from forming in solution, which can get lodged on top of the nanopore. 3.3.4 Preparing DNA suspensions Long DNA molecules, such as λ DNA (N6-methyladenine-free, 500 µg/mL, New Eng- land BioLabs) can be damaged by frequent freezing and thawing cycles, and by the uidic shear forces that arise during pipetting. To minimize damage to the DNA, we carefully separate the purchased stock, using a pipette tip with a wide orice, into 25 µL aliquots, and store them in the fridge at 4◦ C. Prior to a translocation experiment we add 1 mL of degassed 1M KCl solution to the λ DNA stock and heat the solution at 65◦ C for 10 minutes. This heat treatment melts the complementary sticky single-stranded ends of the λ DNA so the molecules become linear, rather than circular or concatenated. We immediately place the micro-centrifuge tube in 31 CHAPTER 3. METHODS FOR NANOPORE EXPERIMENTS an ice bath for 5 minutes, which cools the λ DNA quickly, preventing the ends from reconnecting. 3.3.5 Assembling the nanopore chip holder Figure 3.3.1 (j) shows the interior schematics of the custom-made uidic cell; the nanopore chip is held between two blocks, each of which features uidic reservoirs and ports for a uid inlet, a uid outlet, and Ag/AgCl electrodes. The nanopore chip sits in a recessed seat between two Viton O-rings that hermetically seal the uidic cell when the blocks are held together by machine screws (the schematic does not show ports for the screws). Fluid is introduced into the cell via tubing and driven by syringes attached to shut-o valves. Figures 3.3.1 (h, i) show the nanopore chip holder, made of Peek, with all the uidic lines attached, sitting on Plexiglas alignment box that we use for assembling the nanopore chip in the Peek holder. The assembly starts by having the lower part of the Peek holder sit facing up on the Plexiglas alignment box, best shown in Fig. 3.3.1 (e), with all the inlet and outlet lines attached. Figure 3.3.1 (a) shows a recessed seat for nanopore chip in the lower part of the Peek holder. We align the Viton O-ring with the uid inlet hole, as shown in Fig. 3.3.1 (b). We do this using an optical microscope (5x-10x magnication) to ensure good alignment and that no impurities or bubbles are collected in the inlet during the assembly. We then create a droplet of uid in the middle of the Viton O- ring by slowly pushing uid out using the inlet syringe and then close the inlet valve to hold the droplet in place. This assembly procedure ensures that no air bubbles are entrained between the uid inlet and the nanopore which is placed onto its seat, at side down, as shown in Fig. 3.3.1 (c). The square membrane with embedded nanopore is then aligned with the hole in the Viton O-ring. We then place the second 32 CHAPTER 3. METHODS FOR NANOPORE EXPERIMENTS O-ring on top of the nanopore chip and align it with the square membrane, as shown in Fig. 3.3.1 (d). Some practice is required to lower the top part of the chip holder onto the bottom part, as shown in Figs. 3.3.1 (e,f ), keeping both halves parallel with the Plexiglas alignment box. In Fig. 3.3.1 (g) we inserted the screws that lock the assembly together while holding the top part of the holder to prevent it from wobbling. We tighten the screws slowly to nger-tight, while balancing the tightness of diagonal pairs to maintain the halves parallel. If we over-tighten the screws, a crunch can usually be heard as we break the nanochip. While continuously ushing 10 mM Tris-Cl buer through the assembly we unscrew electrodes a bit from the electrode port to ensure that no air bubbles become trapped between the electrodes and the solution inside the chip holder. If the bubbles get trapped, our signal displays short spikes in ionic current. Repeated ushing with unscrewed electrodes usually solves this issue. We have noticed that DI water slowly etches SiN, so we try to use nanopores in an experiment within a day after they were cleaned and assembled. In our fabrication and assembly procedure, we have especially considered ways of reducing the capacitance, as high capacitance interferes with the usability of the ping-pong technique, described in Ch.5. The capacitance is inversely proportional to the thickness of the insulating material on the at side of the nanopore chip and directly proportional to the area of the at side of the nanopore chip exposed to uid. To increase the thickness of the insulating material on the at side we have fabricated chips with a 3 µm-thick SiO2 layer, instead of 400 nm, and used thicker O-rings. To reduce the area exposed to the uid, we developed a semi-wet assembly procedure. We quickly blow the at side of the nanopore chip dry and place it in the nanopore seat of the holder, at side up instead of at side down (to keep it 33 CHAPTER 3. METHODS FOR NANOPORE EXPERIMENTS Figure 3.3.1: Assembling the nanopore chip holder. (a) A recessed seat for nanopore chip with the uid inlet hole in the lower part of the Peek holder. (b) Aligned Viton O-ring with the uid inlet hole, with a small droplet of uid in the middle of the Viton O-ring. (c) The nanopore chip placed at side down into its seat (the square membrane is aligned with the hole in the Viton O-ring). (d) The second O-ring aligned with the square membrane of the nanopore chip. (e) The lower part of the Peek holder sitting on the Plexiglas alignment box with uidic lines attached. (f ) The top part of the chip holder lowered onto the bottom part. (g) The screws that lock the assembly together are inserted. (h) Peek holder assembled, ready to use. (i) The nanopore chip holder with electrodes and uidic lines attached: inlet lines with syringes and shut-o valves and outlet lines with shut-o valves. (j) The interior schematics of the uidic cell. 34 CHAPTER 3. METHODS FOR NANOPORE EXPERIMENTS dry), and cover it with an O-ring of a smaller inner diameter. The net eect of these modications in fabrication and the assembly is a ve-fold decrease in capacitance, from 300 pF to about 60 pF. 3.4 Recording DNA translocations through a solid- state nanopore We begin the translocation experiment by ushing the uidic lines with degassed 1M KCl solution and placing the nanopore chip assembly and the headstage of the Axon Axopatch 200B inside the Faraday cage (which is grounded by connecting it to the ground plug found at the back of the Axopatch). We ensure that the vibration isolation table on which the Faraday cage sits is properly set up to eectively damp mechanical vibrations. Figure 3.4.1: Characterizing DNA by its translocation through a nanopore. (a) Schematic of a DNA translocation experiment, showing the nanopore chip sandwiched between two reservoirs lled with saline solution. The negatively charged DNA is introduced into the lower, negatively charged reservoir, and is pulled through the nanopore by the electrophoretic force. (b) Five typical DNA translocation events, selected from recordings of a current trace. These electrical signatures reveal the length and folding conformation of the molecule. The gure is from [18]. 35 CHAPTER 3. METHODS FOR NANOPORE EXPERIMENTS We then connect the sockets on the Axopatch headstage to the two Ag/AgCl electrodes on either side of the nanopore assembly. The ground of the headstage should be connected to the lower reservoir (facing the at surface of the nanopore chip), while the input of the headstage should be connected to the upper reservoir. These connections apply a voltage across the nanopore and measure currents through it. Figure 3.4.1 (a) shows a schematic of the DNA translocation setup. We also set the internal Bessel lter of the Axopatch to 10 kHz. After we establish a quiet and steady baseline current through the nanopore, whose standard deviation does not exceed 0.01 nA at 100 mV, and therefore would not obscure the DNA translocation signals (that are on the order of 0.1 nA), we prepare the DNA solution. We slowly draw DNA solution into a 3 mL syringe through wide tubing and tap the syringe to remove any air bubbles that may form near the plunger. We close the inlet valve to the lower reservoir (facing the at side of the nanopore chip) and replace the buer-lled tubing (together with its syringe) attached to that valve with the DNA solution-lled tubing with syringe. We open the valve and push on syringe to ush enough DNA solution through the holder to completely replace the uid in the bottom reservoir (about 0.2 mL). We then apply a positive voltage of 100 mV across the membrane using the Axopatch, and record the electronic current through the nanopore. DNA translocation events will appear as transient blockages in current, similar to those shown in Fig. 3.4.1 (b). 3.4.1 Interpretation of measured current through the nanopore We typically measure a few-nA current of mobile ions owing through the nanopore. At high salt concentrations the charge carriers in the solution dominate the ionic current. The current through the nanopore should then scale linearly with the number 36 CHAPTER 3. METHODS FOR NANOPORE EXPERIMENTS of charge carriers [31]: πV d2pore   4σpore I= (µK + µCl )nKCl e + µK (3.4.1) 4 Lpore dpore where V is the applied voltage, dpore diameter of the nanopore and Lpore its thickness, µK = 7.6 × 10−8 m2 /Vs and µCl = 7.9 × 10−8 m2 /Vs mobilities of the charge carriers, nKCl the number density of K or Cl ions, and σpore the surface charge density of the nanopore that depends on the concentration. The rst term comes from the bulk concentration of ions in the solution and the second from the current due to counter-ions screening the surface charge of the nanopore. At nKCl  2σpore /dpore e, the former dominates the latter. If a polymer passes through the nanopore, it changes the ionic current. This change can either be an enhancement to the signal, or a blockade. The sign and magnitude of current changes are due to two competing eects: an enhancement of the counter-ion current is proportional to the eective charge density on the polymer, and the reduction depends on the conductivity of the solution and the cross sectional area of the polymer which physically blocks the pathway through nanopore [31]. Al- ternatively said, the DNA aects the current through the nanopore by occupying the nanopore and thereby decreasing the number of charge carriers available for bulk ionic transport, but it also adds positive contribution to the ionic current by intro- ducing the positive counter-ions that are shielding its backbone. The enhancements occur at lower salt concentrations, for DNA typically when KCl concentration is be- low 370 mM, as the DNA brings into the nanopore additional charge. The current blockages occur at high salt concentrations, such as 1M. 37 CHAPTER 3. METHODS FOR NANOPORE EXPERIMENTS Figure 3.4.1 (b) shows selection of typical current blockades at 1M KCl. A level of ∆I = 0.3 nA is blocked by a single strand translocating the nanopore, while insertion of two strands simultaneously blocks twice as much current, 2∆I . The change in current through the nanopore due to insertion of a DNA strand can be modeled as [31]: V h π 2 i ∆I = − dDN A (µK + µCl )nKCl e + µK λDN A (3.4.2) Lpore 4 where λDN A is the eective charge on DNA per unit length, and can be taken as a t parameter. Smeets et al. nd that the value of (0.58 ± 0.02) e per basepair ts their data well. However, the Smeets et al. model does not correctly predict experimentally measured current blockades at 1 M KCl using nanopores larger than ∼ 20 nm in diameter [32]. In our experiments, we will be using nanopores of ∼ 10 nm diameter, so the formula given in Eq. 3.4.2 is a good approximation to our data. 38 Chapter 4 The Statistics of DNA Capture by a Nanopore 39 CHAPTER 4. THE STATISTICS OF DNA CAPTURE BY A NANOPORE In this chapter we present a study of DNA translocations of an 8 nm-wide solid- state nanopore which can electrophoretically capture a DNA molecule and pull it through in a folded conguration. The resulting ionic current signals indicate where along their length the DNA molecules were captured. Our study reveals a strongly biased distribution of capture locations, where the probability of capture increases continuously and rapidly towards the DNA's ends. The equilibrium distribution of polymer congurations outside the nanopore oers a natural explanation for this sur- prising nding. We present a simple but successful model of that distribution in which only the congurational entropy is important. We also show that a constant mean translocation velocity and Gaussian velocity uctuations explain the transloca- tion dynamics of folded DNA well, but that a weak length-dependence of the mean segment velocity exists. 4.1 Folded DNA translocations oer clues of DNA capture by a nanopore Most previous studies have focused on instances where the nanopore electrophoret- ically captures DNA at one end and then slides it through in a linear, end-to-end fashion. However, when DNA encounters a ≈ 10 nm-wide solid-state nanopore, the electrophoretic force can initiate translocation anywhere along DNA's length by in- ducing a hairpin fold in the molecule that protrudes into the nanopore [9, 33, 34]. Two segments of DNA extend from the initial fold, a long one of length Ll and a short one of length Ls [Fig. 4.1.1(a)]. The time it takes for each segment to translocate is mea- surable from the time trace of I. Folded DNA translocations entail the simultaneous motion of multiple segments through the nanopore, which may exhibit cooperative 40 CHAPTER 4. THE STATISTICS OF DNA CAPTURE BY A NANOPORE behavior that alters the translocation dynamics [35]. The mechanical bending energy associated with folds may inuence the capture of DNA [36]. Importantly, the study of folded congurations provides snapshots of molecules at the moment of insertion, which oer clues about how the nanopore captures them from solution. Figure 4.1.1: a) A nanopore captures DNA from solution and initiates electrophoretic translocation by forming a hairpin. Segments of length Ll and Ls extend from the capture location. (Detail) TEM image of the 8 nm wide nanopore used. b) Ionic current traces from translocation events of type 1, 2-1, and 2 indicate the capture location. c) The ionic current trace of a folded DNA molecule shows t2 , ttot , and ECD. 41 CHAPTER 4. THE STATISTICS OF DNA CAPTURE BY A NANOPORE 4.2 Brief description of the experimental setup The 8-nm-diameter solid-state nanopore we used [Fig. 4.1.1(a), detail] was fabricated in a 20 nm-thin low-stress silicon nitride membrane following procedures as described in the previous chapter. The nanopore bridged two uid reservoirs containing de- gassed aqueous 1 M KCl, 10 mM Tris-HCl, 1 mM EDTA buer (pH 7.7). An elec- trometer (Axon Axopatch) applied 100 mV across the nanopore and monitored I using two Ag/AgCl electrodes immersed in the reservoirs. A 10 kHz, 8-pole, low-pass Bessel lter conditioned I prior to digitization at 50 kilo-samples per second. The open-pore current was I = 3.6 nA. After adding λ DNA (16.5 µm long, New England Biolabs) to the negatively charged reservoir at a concentration of 24 µg/mL, transient blockages in I were observed, such as the ones shown in Fig. 4.1.1(b). 4.3 Current blockage signals provide snapshots of DNA molecules at the moment of insertion Figure 4.1.1(b) shows illustrations of DNA molecules captured by the nanopore and the corresponding measured current blockage signals. The blockages show quan- tized steps in I that indicate where the nanopore captured each molecule. Unfolded molecules decreased I by ≈ 0.278 nA for the full duration of the translocation event, ttot . We call these type 1 events. Folded molecules cause two segments to occupy the nanopore simultaneously, thereby doubling the reduction in I for a time t2 . Two segments occupied the nanopore for the full duration of type 2 events, indicating molecules captured at the midpoint. A transition from double to single occupancy was observed in type 2-1 events, indicating molecules captured somewhere between 42 CHAPTER 4. THE STATISTICS OF DNA CAPTURE BY A NANOPORE an end and the midpoint. Fig. 4.1.1(c) shows a type 2-1 event that illustrates ttot and t2 ; we judged the occupancy of the nanopore to have changed when I rose or fell 80 % of the way to the next blockage level. In Fig. 4.3.1 we plot a histogram of 20 µs-long samples of current reductions from I caused by translocations of type 1, 2-1 and 2 events. As expected, we observe two distinct equally spaced peaks of current reduc- tions. We also detected event types which indicate molecules captured and folded by the nanopore at multiple locations along its length during the course of translocation. For the present study, however, we restrict our attention to translocations with at most a single fold, which account for ∼ 70% of all events. Figure 4.3.1: Histogram of the number of 20 µs-long current samples per 1 pA bin recorded during type 1, 2-1 and 2 translocation events and the 0.3 ms of baseline that preceded and followed each event. 43 CHAPTER 4. THE STATISTICS OF DNA CAPTURE BY A NANOPORE 4.4 Measured ECD distributions of folded and un- folded DNA molecules We found evidence that a minority of the current blockages were caused by fragments of λ DNA that we wish to exclude from further analysis. We considered the event charge decit (ECD), which is the current blockage integrated over the duration of an event (illustrated in Fig. 4.1.1(c)). Figure 4.4.1 plots the ECD distributions for events of type 1, 2-1 and 2. Most events fall into the main peaks that are centered at 0.408 ± 0.003 pC, regardless of the event type. We attribute those events to intact λ- DNA molecules [33]. Minor peaks in the distributions near 0.15 pC likely correspond to fragments of those molecules. To select a monodisperse ensemble, we excluded events with ECD < 0.27 pC from further analysis. We also excluded six events with ECD> 3 pC, presumably caused by molecules that stuck to the nanopore. These restrictions leave us with an ensemble of ∼ 1100 identical λ DNA molecules that translocated with at most a single fold. 4.5 Translocation dynamics of folded DNA molecules For unfolded molecules the translocation speed, v, can vary in two distinct ways: 1) the speed of a strand can uctuate randomly about the average, and 2) there can be a systematic dependence of the average speed on the initial segment length, eg. t ∝ Lα [11, 13]. Folded DNA molecules have two segments that simultaneously translocate the nanopore, which may aect translocation dynamics [35]. It is not clear whether both segments of a captured DNA molecule, a short one of length Ls and a long one of length Ll , will translocate at the same constant speed, or if the short DNA segment 44 CHAPTER 4. THE STATISTICS OF DNA CAPTURE BY A NANOPORE Figure 4.4.1: Overlaid ECD distributions for translocations of type 1 (dark grey), 2-1 (white), and 2 (medium grey). Events with ECD < 0.27 pC and six with ECD >3 pC were dropped from subsequent analyses in order to exclude fragmented and stuck DNA molecules, respectively. Only 0 pC ≤ ECD ≤ 1 pC is plotted for clarity. will translocate faster than the long segment. Here, we investigate both the inuence of uctuations in v and the length of a segment on the translocation dynamics of folded DNA molecules. 4.5.1 Measured distributions of translocation times We investigate the translocation dynamics of folded DNA molecules by dividing the translocation data into 80 µs bins of t2 . For each ht2 i we plot histograms of ttot , shown in Fig. 4.5.1. We calculate httot i and its standard deviation and plot against ht2 i as shown in Fig. 4.5.2. hQi denotes the mean of quantity Q in a 80 µs bin. If both segments translocated at the same speed, we would expect httot i to decrease in proportion with any increase in ht2 i. Fig. 4.5.2 shows that httot i in fact decreased 45 CHAPTER 4. THE STATISTICS OF DNA CAPTURE BY A NANOPORE approximately linearly with ht2 i until ht2 i ≈ 0.7 ms, where httot i began to rise. That turning point coincides approximately with the mean translocation time for type 2 events. Figure 4.5.1: (a-l) Distributions of ttot for events binned by t2 . The value of ht2 i for each bin is indicated. a) The black curve is a t of Eq. 6 the data, which obtained the values for v0 and ∆v0 /v0 used to generate the curves in (b-l). 4.5.2 Modeling uctuations in the translocation speed Fluctuations in the translocation speed explain the upswing in httot i with ht2 i in Fig. 4.5.2 as the following dynamical model illustrates. Consider a folded molecule whose two segments translocate with the same Gaussian distribution of speeds, Gv0 ,∆v (v); where v0 is mean translocation speed and ∆v is the standard deviation, which ac- counts for uctuations. Accordingly, if a segment translocates in a time t2 , the proba- bility that its length was between Ls and Ls + dLs can be obtained from Gv0 ,∆v (v)dv , 46 CHAPTER 4. THE STATISTICS OF DNA CAPTURE BY A NANOPORE Figure 4.5.2: Dependence of httot i on ht2 i. Error bars indicate the standard deviation of the mean in a 80 µs bin. Bins with ht2 i > 1 ms contain an insignicant number of events (≤ 2). The solid line shows the predictions of the dynamical model that includes velocity uctuations described in text.The dashed line accounts for the α length-dependence of the translocation speed of each segment with t ∝ L . The scaling exponent α = 1.19 ± 0.04 was obtained from a weighted least squares t to the data in the range ht2 i < 0.7 ms. where v = Ls /t2 :   Ls dLs P (Ls | t2 ) dLs ∝ Gv0 ,∆v0 . (4.5.1) t2 t2 The probability distribution P (Ls | t2 ) dLs is normalized by integrating over Ls from 0 to L. The complementary segment has length Ll = L − Ls . The probability that it takes between ttot and ttot + dttot to translocate is:   L − Ls L − Ls P (ttot | Ls ) dttot ∝ Gv0 ,∆v0 dttot . (4.5.2) ttot t2tot 47 CHAPTER 4. THE STATISTICS OF DNA CAPTURE BY A NANOPORE Combining Eqs. 4.5.1 and 4.5.2, we nd that when one segment translocates in a time t2 , the complementary segment will translocate in a time between ttot and ttot + dttot with a probability given by: ˆ L  P (ttot | t2 ) dttot ∝ P (ttot | Ls ) P (Ls | t2 ) dLs dttot . (4.5.3) 0 The distribution P (ttot | t2 ) is normalized by integrating over ttot from t2 to ∞. A least squares t of Eq. 4.5.3 to the ttot histogram shown in Fig. 4.5.1 (a) obtains v0 = 10.76 ± 0.06 mm/s and ∆v/v0 = 0.198 ± 0.005. With those parameters and Eq. 4.5.3, we predicted the distributions of ttot for each bin of t2 as shown in Fig. 4.5.1 (b-l). All ttot histograms are well described by our model which assumes that both segments translocate with the constant v0 . We used the same parameters and Eq. 4.5.3 to cal- culate httot i as a function of t2 and plotted the results in Fig. 4.5.2. The predicted re- lationship agrees well with the data. These comparisons show that our two-parameter model gives a reasonable description of the dynamics of DNA translocations. 4.5.3 Possible length-dependence on translocation dynamics Long molecules are known to translocate more slowly than short ones in unfolded congurations [11] because the moving segment is longer and experiences more viscous drag when it is drawn to the nanopore from a large coil [12, 13]. Storm et al. assumed a power law relationship between the translocation time and the length of unfolded DNA, Ttr ∝ Lα , and found the scaling exponent α = 1.27 [9]. Assuming each segment of a folded molecule obeys a similar scaling relationship and using Ls + Ll = L, we 1/α 1/α nd ttot = (Ttr − t2 )α . We tted that expression to the data in Fig. 4.5.2 for ht2 i < 0.7 ms to obtain α = 1.19 ± 0.04. The slope of the data in Fig. 4.5.2 for 48 CHAPTER 4. THE STATISTICS OF DNA CAPTURE BY A NANOPORE ht2 i < 0.7 ms reveals a weak systematic dependence of the translocation speed on the length of a segment. 4.6 The statistics of DNA capture by a nanopore We next turn to mapping distances along DNA molecules. In order to nd where along its length the molecule gets captured by the nanopore, we dene the capture Ls location, x ≡ Ls +Ll , as the fractional contour distance from the initial fold to the nearest end. We measure the time for each segment to translocate from the time trace of I [33, 34, 9] and use these measurements to estimate x. Storm et al. inferred the distribution of x for λ DNA translocations and concluded that folds occur with equal probability everywhere along a molecule's length, but that the DNA is more likely to be captured at its ends because of the lower energetic cost of threading an unfolded molecule [9]. This implies that molecules test multiple congurations prior to capture, which is a statistical process governed by energetic considerations. By contrast, Chen et al. reported a bias for unfolded translocations that increased with applied voltage [34]. This nding implies that molecules pre-align in the elds outside the pore rather than sample multiple congurations prior to capture. No model for the distribution of x is available to help evaluate these competing pictures. 4.6.1 The measured distribution of capture locations For each translocation event, such as the ones shown in Fig. 4.1.1(b), we obtained the capture location, x, by assuming the translocation speed, v, was constant over the 49 CHAPTER 4. THE STATISTICS OF DNA CAPTURE BY A NANOPORE duration of the event, which follows the approach of Storm et al. [9] and gives: t2 x= . (4.6.1) t2 + ttot In the previous section we have shown this assumption to be reasonable. Later, we shall further discuss the inuence of uctuations and a weak contour length depen- dence in v on the distribution of x. Figure 4.6.1: Distribution of capture locations. The stacked histogram bars for translocations of type 1 (dark grey), 2-1 (yellow), and 2 (light grey) indicate the number of events of each type in a bin. Data points indicate the total number of events of all types and their mean x in a bin. Error bars indicate the square root of the total events. The solid line shows the distribution predicted by Eq. 4.6.7 for the S S theoretical γ1 = 0.703 and γ2 = 0.203 [37]. Figure 4.6.1 presents a histogram of the capture locations. We selected a bin size that avoids a possible artifact of the limited measurement bandwidth; since there is 50 CHAPTER 4. THE STATISTICS OF DNA CAPTURE BY A NANOPORE a lower bound on t2 , it would be dicult to populate bins near x=0 if the bin size were too small. The distribution shows that the frequency of capture was highest near x = 0, decreasing rapidly but smoothly with distance away from the ends, and becoming a slowly decreasing function of x near x = 0.5. The bin that includes x = 0.5 rises above the trend. 4.6.2 Modeling the capture location distribution We propose a physical model to explain the distribution of DNA capture locations by a solid-state nanopore. We assume that a DNA molecule has enough time to sample all available congurations as it approaches the nanopore. At the moment of capture, the nanopore randomly selects a conguration from the equilibrium ensemble. We model that conguration as a pair of self-avoiding walks (SAWs) of lengths Ls and Ll , tethered to the surface at a single point representing the nanopore. We use results from Duplantier's theoretical work on polymer networks [38], which showed how the partition function, Z, for an arbitrary polymer network could be factorized in terms of properties of the vertices of the network. Z, which is also called the conguration number in Duplantier's work, is equivalent to Ω in our model. For a single polymer of length L tethered to a surface by one end, such as shown in Fig. 4.6.2 (a), the total number SAWs, Ω(L), has the following asymptotic form [21, 38]: S Ω(L) ∼ µL Lγ1 −1 . (4.6.2) γ1S is a universal surface scaling exponent which depends solely on the dimensionality of the lattice and µ is the lattice coordination number. Barber et al. studied SAWs tethered to a surface and obtained γ1S ≈ 0.70 from simulations on a cubic lattice [39]. 51 CHAPTER 4. THE STATISTICS OF DNA CAPTURE BY A NANOPORE Figure 4.6.2: DNA capture modeled as polymer tethered to the surface. The number of congurations available to a polymer of length L tethered at x, ΩL (x), would simply be Ω(Ls )Ω(Ll ) if both segments behaved independently. In addi- tion to undergoing self-avoiding walks, however, the segments must avoid one another. The exponent γ2S ≈ 0.203 accounts for this restriction in a pair of tethered SAWs of equal length [37]. In the theory of polymer networks, the number of congurations L of two polymers of equal length, , each tethered to a common point on a surface by 2 one end, as shown in Fig. 4.6.2 (b), is given by [38]:    γ2S −1 1 L L ΩL ∼µ . (4.6.3) 2 2 We also interpret Eq. 4.6.3 as the conguration number of a polymer of length L 1 1 < Ω( L2 )2  tethered at its midpoint, x= 2 . In general, ΩL 2 because excluded-volume interactions forbid congurations where the two polymer segments overlap. To obtain the conguration number of a polymer tethered at arbitrary x (i.e. where the two segments tethered to the same point on a surface have dierent lengths), we make use of the theory of contacts in polymer networks [38]. When two points in an arbitrary polymer network are brought into contact and fused permanently together, the conguration number is reduced by a factor proportional to L−Υ , where L is the 52 CHAPTER 4. THE STATISTICS OF DNA CAPTURE BY A NANOPORE length scale characterizing the segments between the vertices of the network, and Υ is the contact exponent, which accounts for the reduction in congurations due to the interaction between segments [38]. Here we consider an extremely simple polymer network comprising only two segments, each tethered to a surface, a shorter one of length Ls and the other of length Ll , as shown in Fig. 4.6.2 (c). Initially, the segments are tethered far enough apart that they do not interact, and the conguration number of the network is Ω(Ls )Ω(Ll ). The tethered ends are then brought into contact and fused. The conguration numbers before and after contacting the ends are related by: ΩLs +Ll (x) = λΩ(Ls )Ω(Ll )L−Υ s , (4.6.4) where λ is a constant. We determine the contact exponent and the proportionality constant by noting that ΩLs +Ll (x) must reduce to Eq. 4.6.3 when length of a polymer is Ls = Ll = L/2 giving:    γ2S −1  2γ1S −2  −Υ 1 L L L L L ΩL ∼µ = λµ . (4.6.5) 2 2 2 2 From Eq. 4.6.5 we nd λ=1 and Υ = 2γ1S − γ2S − 1. Inserting these expressions into Eq. 4.6.4, with Ls + Ll = L, we obtain: S S S ΩL (x) ∼ µL (L − Ls )γ1 −1 Lγs 2 −γ1 . (4.6.6) Note that Eq. 4.6.6 gives the correct scaling in the other limiting case ΩL (0) ∼ S µL Lγ1 −1 (for Ls → 0). The probability of capturing a DNA molecule at x, P (x), is proportional to ΩL (x), so from Eq. 4.6.6 and x ≡ Ls /L with introducing the normalization constant A, we 53 CHAPTER 4. THE STATISTICS OF DNA CAPTURE BY A NANOPORE obtain: S S S P (x) = A(1 − x)γ1 −1 xγ2 −γ1 . (4.6.7) The solid line in Fig. 4.6.1 plots the distribution of capture locations predicted by Eq. 4.6.7. The constant A was obtained from a weighted least squares t to the data. The tethered-polymer model describes the observed distribution of capture locations well. Note that the skewness arises naturally from congurational entropy alone; every DNA conguration is represented with equal probability and there is no need to invoke a bending energy, as Storm et al. did, to explain the preponderance of molecules captured near their ends [9]. The model disagrees most signicantly with the data at x = 0.5, where more events were observed than predicted. That discrepancy can be explained by the translocation of circular λ DNA molecules, whose complementary single-stranded ends had bound, resulting in an excess of type 2 events. An important implication of our model is that DNA does not search for an energetically favorable conguration before initiating a translocation. 4.6.3 On the importance of excluded-volume statistics In Fig. 4.6.1, we plotted the theoretical distribution of capture locations using scal- ing exponents valid for self-avoiding walks, which implies that the DNA exhibited excluded-volume congurational statistics. This assumption is consistent with the available experimental evidence, which nds a Flory exponent consistent with excluded- 1 volume statistics for double-stranded DNA molecules as short as times the length of 30 λ DNA [40, 41, 42, 43]. Those experiments were performed at low salt concentration, however, where the eective diameter of DNA, including electrostatic contributions, 54 CHAPTER 4. THE STATISTICS OF DNA CAPTURE BY A NANOPORE is much larger than was the case in our experiments. A theoretical argument based on scaling theory predicts excluded-volume interactions will only be fully developed when a sucient length is reached, which turns out to be about four times the length of λ DNA at high salt concentrations [44]. In the short-molecule limit, DNA is pre- dicted to exhibit ideal chain statistics, with surface scaling exponents γ1S = 0.5 and γ2S = 0 [38]. Figure 4.6.3: Distributions of capture locations. The stacked histogram bars for translocations of type 1 (dark grey), 2-1 (yellow), and 2 (light grey) indicate the number of events of each type in a bin. Data points indicate the total number of events of all types and their mean x in a bin. Error bars indicate the square root of the total events. The distribution obtained assuming v is constant. The solid line shows the prediction of Eq. 4.6.7 for self-avoiding walks, whose surface scaling S S exponents are γ1 = 0.703 and γ2 = 0.203 [37]. The dashed line shows prediction of S S Eq. 4.6.7 for ideal chains, whose surface scaling exponents are γ1 = 0.5 and γ2 = 0 [38]. 55 CHAPTER 4. THE STATISTICS OF DNA CAPTURE BY A NANOPORE Figure 4.6.3 compares the experimental distribution of DNA capture locations with those predicted for SAWs and for ideal chains. The dierence between SAWs and ideal chains is small. SAW statistics predict a slightly lower capture probability as x approaches 0.5 and t the data slightly better. This comparison makes it clear our assumption that DNA exhibits excluded-volume statistics is not crucial. 4.6.4 Inuence of a length-dependent translocation speed on the distribution of capture locations Figure 4.5.2 revealed a weak dependence of v of a segment on its initial length. We used the power law scaling relationship, t ∝ Lα , to relate the translocation time to the initial length of a segment and obtained α = 1.19 from a t to the data. It follows 1/α t2 that the capture location for each event is x= 1/α 1/α . We plot the distribution of t2 +ttot capture locations in Fig. 4.6.4, using a larger bin size chosen to avoid an artifact as described before. The distribution still reveals a strong bias for capturing the DNA close to its end. The capture probability rapidly decreases with distance away from the ends. The prediction of our capture location model for SAWs, Eq. 4.6.7, agrees well with the data. We conclude that even though the shorter strand translocates somewhat faster on average, the impact on the distribution of x is too slight to aect its agreement with our model. 4.6.5 Inuence of translocation speed uctuations on the dis- tribution of capture locations Fluctuations in v can cause the short segment of a folded DNA molecule to translocate faster than the long segment, leading to an overestimate of x. Can such random 56 CHAPTER 4. THE STATISTICS OF DNA CAPTURE BY A NANOPORE Figure 4.6.4: Distributions of capture locations. The stacked histogram bars for translocations of type 1 (dark grey), 2-1 (yellow), and 2 (light grey) indicate the number of events of each type in a bin. Data points indicate the total number of events of all types and their mean x in a bin. Error bars indicate the square root of the total events. The number of bins reects experimental bandwidth limitations. The distribution obtained assuming a length-dependent v characterized by α = 1.19. The solid line shows the prediction of Eq. 4.6.7 for self-avoiding walks. uctuations distort the distribution of capture locations and create an articial bias for capturing DNA near its ends? We addressed this question by studying a model of DNA translocation dynamics with a Gaussian distribution of translocation speeds. Here we show that uctuations in v have only a minor inuence on the distribution of capture locations. When the mean translocation speed is constant, ttot is related 1−x  to t2 and x by ttot = t2 x . That expression can be combined with the expression 1−x t2   for P (ttot | t2 ) dttot in Eq. 4.5.3 to obtain P t2 x | t2 x2 dx, the probability that 57 CHAPTER 4. THE STATISTICS OF DNA CAPTURE BY A NANOPORE the capture location is between x and x + dx for a given t2 . Hence, if the measured distribution of t2 values is f (t2 ), the resulting distribution of capture locations is given by: ˆ ∞     1−x t2 P (x) ∝ P t2 | t2 f (t2 )dt2 . (4.6.8) 0 x x2 The distribution P (x) is normalized by integrating over x between 0 and 1/2. Figure 4.6.5: Distribution of capture locations predicted for a uniform distribution of t2 between 0 and L/2v0 and including the eects of translocation speed uctuations. The curve was obtained by evaluating Eq. 4.6.7 numerically. Figure 4.6.5 plots the distribution P (x) for a uniform distribution of t2 values (i.e. f (t2 ) is constant) between 0 and L/2v0 , and 0 otherwise. We would expect to observe such a distribution in the absence of translocation speed uctuations and if DNA were captured with equal probability everywhere along its length. Since P (x) is very 58 CHAPTER 4. THE STATISTICS OF DNA CAPTURE BY A NANOPORE at for x < 0.4, the highly skewed distribution of x we measured experimentally [Fig. 4.6.1] could not have resulted from uctuations in the translocation speed; the observed bias in capture locations is real. The dip in P (x) near x = 0.5 results from the unnaturally abrupt distribution of f (t2 ) we chose. In the presence of translocation speed uctuations, the actual range of t2 extends beyond L/2v0 and attens P (x) near x = 0.5, as we measured. 4.7 Discussion A question that our experiments cannot address is where, in relation to the nanopore, the capture location is determined. Within our model, x is determined at the nanopore; however, recent studies have identied a critical radius from the nanopore, typically on the scale of hundreds of nanometers, within which electrophoretic forces overwhelm diusion [8, 45]. It is possible that the rst segment to insert is transported essentially deterministically to the nanopore from some distance away without alter- ing the distribution of x. Similarly, our assumption that a molecule is at equilibrium prior to capture is not seriously compromised if it becomes stretched out of equi- librium by the eld gradients only after the capture location has been determined. Alternatively, the forces on DNA beyond the nanopore may restrict the available congurations and thereby reduce γ1S and γ2S . The dynamical model of the distribution of translocation times of Sec. 4.5, which provides agreement with data in Fig. 4.5.2 by using an assumption of constant translo- cation speed in Eq. 4.5.3, enables obtaining the distribution of x in Fig. 4.6.1. Fluc- tuations lead to errors in estimating x for a particular event, as one segment may translocate faster than the other, but the relationship between ttot and t2 is the same 59 CHAPTER 4. THE STATISTICS OF DNA CAPTURE BY A NANOPORE on average as if v were constant. Accordingly, the model predicts uctuations in v have only a minor inuence on P (x). We note that events with t2 > 0.7 ms are drawn from tails of the speed distributions; httot i rises with ht2 i because both segments of molecules captured at x ≈ 0.5 translocated more slowly than average during t2 , not because the segments translocated at dierent speeds on average. 4.8 Conclusion We measured the distribution of capture locations along λ DNA molecules by an 8 nm wide solid-state nanopore and presented a theoretical model which explains that distribution. Surprisingly, the strong bias for capturing molecules near their ends is a consequence of the congurational entropy of the approaching polymer; molecules do not search for an energetically favorable conguration before translocating. We also used folded DNA congurations to probe the dynamics of multiple polymer segments translocating a nanopore simultaneously, thereby quantifying the uctuations and the length dependence of the translocation speed. 60 Chapter 5 Non-Equilibrium Dynamics of λ DNA Probed by the Molecular Ping-Pong Technique 61 CHAPTER 5. NON-EQUILIBRIUM DYNAMICS OF λ DNA PROBED BY THE MOLECULAR PING-PONG TECHNIQUE In this chapter we explore the relaxation dynamics of the DNA from a compressed non-equilibrium conguration by using the delayed capture and re-capture, so-called molecular ping-pong technique [8]. In DNA ping-pong, a single molecule is shued back and forth through a nanopore by reversing the applied voltage after each translo- cation. The fast translocation process drives the DNA into a compressed state and the subsequent recapture quanties its re-equilibration via the ionic current blockade signal. Since the re-equilibration time of λ DNA is longer than the intervals between subsequent voltage reversals, the DNA can be recaptured in dierent stages of relax- ation, which is revealed through the measurements of event-charge-decits, current blockades and recapture times. 62 CHAPTER 5. NON-EQUILIBRIUM DYNAMICS OF λ DNA PROBED BY THE MOLECULAR PING-PONG TECHNIQUE 5.1 Nanopores can probe non-equilibrium congura- tions of DNA coils Early nanopore experiments found that the mean time it takes a molecule to translo- cate a nanopore end-to-end, Ttr , increases with DNA's contour length, L, in a super- linear fashion, Ttr ∝ L1.27 [10, 11]. This result has been attributed to the inuence of the viscous drag, acting on the length of DNA polymer that is pulled through solution towards and into the nanopore under the inuence of electrophoretic force [12, 14]. A longer DNA, which forms a larger coil in solution, will experience more viscous drag and translocate more slowly than a shorter one because its segments are pulled from on average longer distances into the nanopore. For the same reason, identical DNA molecules that initiate translocation in extended congurations will translocate more slowly than the ones in more compact congurations closer to the nanopore, giving an explanation for the experimentally observed large uctuations in Ttr [13]. These results indicate that molecules translocate at an average speed that reects their initial conguration, which creates an opportunity to rapidly study the congurations of DNA molecules, including those driven far from equilibrium. The translocation process itself drives long DNA molecules, such as λ DNA, into a compressed, out-of-equilibrium state. The electrophoretic driving force acts on the segment of DNA inside the nanopore and delivers the DNA to the receiving reservoir in ∼ 2 ms, a time that is nearly two orders of magnitude shorter than the polymer's longest relaxation time [46]. Indeed, simulations showed that DNA is compressed after translocation [47]. To probe the conguration of the DNA molecule one can use a novel technique called molecular ping-pong, which enables recapturing the same molecule by reversing the driving voltage after the DNA translocates the nanopore. 63 CHAPTER 5. NON-EQUILIBRIUM DYNAMICS OF λ DNA PROBED BY THE MOLECULAR PING-PONG TECHNIQUE In this way, Plesa et al. [48] measured folding of a single DNA hundreds of times. They found high degrees of folding when the DNA was recaptured shortly after the translocation and they observed that molecule's folding decreased with increasing re- capture time, tR , demonstrating that nanopores can measure re-equilibration of DNA. Since a more compressed molecule oers multiple segments for simultaneous capture by the nanopore, they inferred that the DNA is compressed after the translocation and relaxes on its way back. However, they chose to measure folding, for which no dynamical model is available, and so it is still unknown how does the decrease in folding relate to the DNA coil expansion. Furthermore, the data they plot does not take into account that the DNA spent some time relaxing even before the voltage reversal, so their measurements cannot be used to quantify DNA relaxation. Here, we show that nanopores and the molecular ping-pong technique are sensi- tive enough to rapidly characterize the relaxation of λ DNA molecules. We indirectly measure relaxation of DNA molecules through three independent quantities, by mea- suring the recapture times, current reductions and average translocation speed. We nd that the recaptures of the DNA are hastened by its expansion, as its trailing segments will on average be closer to the nanopore at the moment of voltage reversal. Similarly to [48], we also nd DNA molecules translocate through the nanopore with more folds as segments in a more compressed coil occupy smaller volume and are more likely to be be simultaneously captured and pulled into the nanopore. Finally, the more compressed DNA translocates faster as the uidic drag on segments pulled into the nanopore from a smaller coil is on average smaller. The measurements of recapture times, increase in folding and translocation speed together contribute to characterization of the state the DNA is in at any point in time after a translocation. By experimentally controlling tD , the time between completion of the translocation 64 CHAPTER 5. NON-EQUILIBRIUM DYNAMICS OF λ DNA PROBED BY THE MOLECULAR PING-PONG TECHNIQUE and reversal of the driving voltage, we draw molecules back to the nanopore at various times to take snapshots of their initial congurations. A simple model of DNA coil expansion oers a natural explanation for our observations and provides an excellent t to our translocation speed measurements. 5.2 A short description of the experimental setup We probed the out-of-equilibrium relaxation dynamics of long λ DNA biopolymers by three dierent nanopores, a 9 × 10 nm-wide solid-state nanopore A [Fig. 5.2.1(a), detail], a 9 × 7.7 nm-wide nanopore B, and 8.7 × 7.6 nm-wide nanopore C. For nanopore A, we fabricated the free-standing SiN membrane suspended on a stack of two 400-nm-thick layers of SiO2 and SiN, following procedure described in Ch. 3. The two other devices had a 3µm-thick oxide layer, which reduced the capacitance of the nanopore device to 60 pF. Each nanopore joined two ionic solution reservoirs containing 1 M KCl (pH 8) in deionized water. A voltage of +100 mV, applied by an Axon Axopatch electrom- eter via Ag/AgCl electrodes induced 13.8 nA, 8.3 nA and 11.1 nA ionic current I through nanopores A, B and C, respectively. We conditioned the signal by a 10 kHz, low-pass Bessel lter before digitization at 250 kilo-samples per second and then recorded I using the electrometer. After introducing a 11 µg/mL DNA solution into the negatively-charged reservoir, we observed transient blockages of I, such as those shown in Fig. 5.2.1 (b). Molecular ping-pong experiment was achieved via a LabView- congured FPGA (Field-Programmable Gate Array) card, which by monitoring I can sense the DNA and then reverse the polarity of driving voltage after delay time, tD , which can be adjusted arbitrarily. 65 CHAPTER 5. NON-EQUILIBRIUM DYNAMICS OF λ DNA PROBED BY THE MOLECULAR PING-PONG TECHNIQUE Figure 5.2.1: (a) Illustration of the molecular ping-pong probing the relaxation of DNA after translocation. Detail: TEM image of a ∼ 10 nm-wide nanopore. (b) Ionic λ DNA. (c) Translocation events current trace of 5 successive translocations of a single of the ping-pong sequence in (b) after the baseline correction. We measure ∆I and ECD with respect to the average of a 0.2 ms-long baseline after each event. 66 CHAPTER 5. NON-EQUILIBRIUM DYNAMICS OF λ DNA PROBED BY THE MOLECULAR PING-PONG TECHNIQUE 5.3 Current trace of a ping-pong sequence Figure 5.2.1 (a) illustrates the ping-pong experiment and Fig. 5.2.1 (b) shows a typical time recording of I on nanopore A. Voltage of +100 mV applied across the nanopore drives the λ DNA molecule to translocate the nanopore, which we observe as a tran- sient blockage of I [33]. Our software recognizes translocation when the current drops from the baseline below the xed threshold Ith = 13.75 nA for at least 0.3 ms. After the blockade ends, we continue to apply +100 mV for tD = 5 ms. We then reverse the voltage bias. Following the voltage reversal, the molecule is drawn back to the nanopore and recaptured in time tR , which we observe as another disruption of I. The cycle of delayed voltage reversals and recaptures is repeated several times, inducing the molecule in a compressed state and allowing it to relax for tD + tR , which is the time we measured between successive translocations. In our experiment we controlled tD and in Fig. 5.2.1(b) it is equal to 5 ms following each translocation. We label the initial capture of a molecule in a ping-pong sequence with N = 1, its rst recapture (the 2nd translocation) with N =2 and so on. We chose to recapture a single molecule for up to N = 27 while waiting at most 5tD (which is 25 ms in the example given in Fig. 5.2.1(b)) after each voltage reversal for its recapture. After N = 27 or if the DNA was not recaptured, we applied +100 mV for 1 second to drive this molecule away from the nanopore and then allowed another molecule to start a new ping-pong sequence. In our experiment we captured ∼ 100 molecules per tD , which we varied from 2 to 150 ms. Nanopores A, B and C probed ∼ 1200, ∼ 830 and ∼ 600 molecules, respectively. 67 CHAPTER 5. NON-EQUILIBRIUM DYNAMICS OF λ DNA PROBED BY THE MOLECULAR PING-PONG TECHNIQUE 5.4 A brief description of data analysis Rapid reversals of voltage bias that enable DNA recapture also cause spikes in ionic current, shown in Fig. 5.2.1(b), due to capacitative charging of the membrane, as I ∝ CdV /dt. The current settles to its steady-state value in a time that is inuenced by the capacitance of the nanopore chip. Translocation signals of DNA molecules will appear on the decaying slope of I if DNA is recaptured shortly after the voltage reversal. This usually happens for short tD . In order to analyze the translocation signals and take measurements, we rst have to correct for the slope of I between voltage reversals. Performing the analysis on the absolute value of I, we t one chosen decaying slope of I to a polynomial function. The t is then subtracted from the current traces following each voltage reversal [for details see Appendix]. Figure 5.2.1(c) shows ∼ 1ms-long at-bottomed current blockades events recovered from Fig. 5.2.1(b) after baseline subtraction. 5.5 Experimental clues of DNA relaxation from a compressed state Experimental clues of how λ DNA coil relaxes after translocation can be obtained by comparing measurements on N = 1 translocations, which we would collect in regular translocation experiments, to those of molecules' recaptures for various choices of tD . We measured tR , current reductions ∆I , and the integrated area under the current blockade signal, called ECD (event-charge-decit). The quantized ∆I levels, for instance `1-3-1' in the N =3 event of Fig. 5.2.1 (c), reveal the number of folded 68 CHAPTER 5. NON-EQUILIBRIUM DYNAMICS OF λ DNA PROBED BY THE MOLECULAR PING-PONG TECHNIQUE segments that simultaneously occupy the nanopore during translocation [33, 49]. ECD measured for folded events, such as the N = 3, would be the same even if the molecule translocated end-to-end [10, 19], for which a simple relation applies [33, 34, 49], ECD = Ttr ∆I (5.5.1) This enables us to use ECD to calculate the average translocation speed, v = L/Ttr , for all identical molecules. 5.5.1 ECD distributions of the rst capture of DNA by the nanopore Since the relaxation of DNA is a length-dependent process [21], we want to make sure that our analysis includes only molecules of the same length. Figure 5.5.1 plots the histogram of ECD for N =1 translocations, ECD1 , collected by nanopores A, B and C. For all three nanopores, we observe that most events distribute around the main peak centered at ∼ 0.3 pC and we attribute those to intact λ DNA. For nanopore A, we attribute events with ECD1 < 0.2 pC to fragments of a long λ DNA molecule and events with ECD1 > 0.44 pC to either concatenated or DNA stuck to the nanopore during translocation. These lower and upper bounds are chosen as departures of 1.5 standard deviations below the histogram peak and of 2.5 standard deviations above the histogram peak, and are shown in Fig. 5.5.1 as red dotted lines for all three nanopores. In our analysis we include only events within these two bounds, leaving us with 70 − 80% of molecules per nanopore. 69 CHAPTER 5. NON-EQUILIBRIUM DYNAMICS OF λ DNA PROBED BY THE MOLECULAR PING-PONG TECHNIQUE Figure 5.5.1: ECD1 histograms of captured molecules, obtained on nanopores A, B and C. For clarity we show only ECD1 < 0.8 pC. The dotted lines correspond to departures of 1.5 standard deviations below the histogram peak and of 2.5 standard deviations above the histogram peak, obtained from a Gaussian t to the main peak. 5.5.2 Measured distributions of recapture times Figure 5.5.2 plots tR histograms, showing how long it takes for identical molecules to be recaptured after they have been driven away from the nanopore for tD ranging between 5 and 150 ms. All tR histograms feature a peak and a tail that extends across the whole 5tD range that we waited for recapture in our experiment. For tD = 5 ms the majority of recaptures populates a peak centered at ∼ 1 ms, with a very small fraction of molecules returning to the nanopore in time longer than 5 ms. As we increase tD , the fraction of molecules returning to the nanopore in times longer than the chosen tD increases. Recaptures start spanning the whole 5tD range, peaks of tR histograms shift towards later tR values and the time it takes for molecules' recapture start varying more. We nd the recapture time distributions strongly depend on the choice of tD . 5.5.2.1 Modeling recapture time distributions We use the drift-diusion model given with the Smoluchowski equation to calculate how long would it take a structureless particle to return to the nanopore after being 70 CHAPTER 5. NON-EQUILIBRIUM DYNAMICS OF λ DNA PROBED BY THE MOLECULAR PING-PONG TECHNIQUE Figure 5.5.2: Histograms of tR measurements for tD in the range between 5 and 150 ms, binned in 0.1 tD -wide bins collected on nanopore A. Solid red lines are predictions of the drift-diusion model. driven away for time tD . Gershow and Golovchenko used this model to predict recap- ture times of short, 4− and 6−kbp DNA molecules and found they agreed well with their experimentally measured tR histograms [8]. Here we use the same approach to model recapture rates of a ∼ 10 times longer λ DNA. We assume the structureless particle suspended in a solution of conductivity σ = 10.25 (Ωm)−1 has electrophoretic mobility µ = 4.2 × 10−4 cm2 /Vs and diu- sion constant D = 0.588 µm2 /s, which are the values measured for λ DNA [50]. We describe the motion of the particle outside the nanopore with the Smoluchowski equation that accounts for drift in the electric eld in the vicinity of the nanopore, E = ∓|I|/(2πσr2 ) [shown in Fig. 5.5.3 (a)], and diusion due to random collisions with the solvent molecules [details of the theoretical model are presented in the Appendix]. 71 CHAPTER 5. NON-EQUILIBRIUM DYNAMICS OF λ DNA PROBED BY THE MOLECULAR PING-PONG TECHNIQUE As shown in Fig. 5.5.3 (b), the translocated molecule is taken to be initially located about 100 nm away from the nanopore, and the probability density of the initial center of mass location is described with a delta function. We numerically evolve the initial condition for time tD , imposing the no-ux boundary conditions at zero and at innity (which numerically corresponds to choosing some large distance away from the nanopore). We observe that the peak of the probability density of the particle's position drifts away from the nanopore with increasing tD , while the widening of the distribution is a consequence of random diusive forces. We solve for the particle's motion back towards the nanopore by setting the so- lution obtained at time t = tD as the initial probability density of particle's position at the moment of voltage reversal. We impose the absorbing boundary condition at the nanopore and propagate the initial probability density for time 5tD . Fig- ures 5.5.3 (c,d) show the numerical solutions for motion towards the nanopore, for the choice of tD = 5 ms and tD = 100 ms. In both gures, the slight motion of the peak towards the nanopore is due to the electrophoretic drift, whereas the motion of the peak away at later times is indicative of the part of the distribution being absorbed by the nanopore. Finally, we construct the recapture rates [see Appendix for details] and show them in Fig. 5.5.3(e). The drift-diusion model predicts that molecules will most likely be recaptured at tR ' tD . 5.5.2.2 Comparing predictions of the drift-diusion model to experimen- tal data Figure 5.5.2 compares theoretically predicted recapture rates for a structureless parti- cle (red solid lines) to experimentally measured recaptures of λ DNA. The rates were normalized to their most probable values and laid over measured tR histograms. All 72 CHAPTER 5. NON-EQUILIBRIUM DYNAMICS OF λ DNA PROBED BY THE MOLECULAR PING-PONG TECHNIQUE Figure 5.5.3: Numerical solutions of the drift-diusion model. (a) Electric eld in the 2 vicinity of the nanopore. (b) Evolution of the probability 2πr c(r, t) for the motion of DNA away from the nanopore. The narrow red strip at t = 0 ms corresponds to the initial condition. (c) Numerical solution for the motion towards the nanopore after tD = 5 ms away. (d) Numerical solution for the motion towards the nanopore after tD = 100 ms away. (e) Recapture rates for tD ranging from 5 to 150 ms. 73 CHAPTER 5. NON-EQUILIBRIUM DYNAMICS OF λ DNA PROBED BY THE MOLECULAR PING-PONG TECHNIQUE theoretically predicted recapture rates feature a peak centered at tR ' tD . For short tD = 5 ms, the theory grossly overestimates the time needed for recapture of λ DNA. As we increase tD , the discrepancy between the measured and theoretically predicted recapture distributions decreases. For tD = 150 ms we nd the theoretically obtained recapture distribution compares to data rather well. In order to evaluate the discrepancy between recapture rates predicted for a parti- cle and ones measured for λ DNA, in Fig. 5.5.4 we plot the relative dierence between the model and histogram mean, (htR imodel − htR i)/htR imodel , as a function of tD . For tD = 2 ms, we nd the DNA was recaptured ∼ 80% faster than predicted for a struc- truless particle. As we increase tD , the relative dierence between the model and the data steadily decreases, reaching ∼ 20% dierence for tD = 150ms, and sug- gesting that recaptures of DNA at late times can well be described with those of a structureless particle. Figure 5.5.4: Relative dierence between the mean of recapture rates obtained from the drift-diusion model and tR histogram mean, taken over 5tD range. Error bars (some of which are within the size of a symbol) represent standard deviations of the mean for each tR histogram. 74 CHAPTER 5. NON-EQUILIBRIUM DYNAMICS OF λ DNA PROBED BY THE MOLECULAR PING-PONG TECHNIQUE 5.5.2.3 Recapture time distributions suggest expansion of DNA from an initially compressed state The initial expansion of DNA coil may account for fast recaptures for short tD . Illus- tration in Fig. 5.5.5 shows that, unlike the point particle, as DNA drifts away from the nanopore, it also expands. As a consequence, some segments of the DNA coil are likely to linger closer to the nanopore. As we reverse the voltage, the segments closest to the nanopore get captured rst, resulting in hastened recaptures. Figure 5.5.5: Illustration comparing the motion of DNA and structureless particle away from the nanopore. (a) DNA, which forms a random coil in solution, expands while it moves away from the nanopore. Upon voltage reversal the segments closest to the nanopore get captured rst, resulting in hastened recaptures. (b) A point particle moves away and towards the nanopore in comparable time. 5.5.3 Frequent recaptures of molecules with multiple folds sug- gest more compressed DNA congurations Figure 5.5.6 (a) compares histograms of 4 µs-long samples of ∆I obtained from N >1 translocation signals on nanopore A for tD = 5 ms and tD = 100 ms. Both histograms 75 CHAPTER 5. NON-EQUILIBRIUM DYNAMICS OF λ DNA PROBED BY THE MOLECULAR PING-PONG TECHNIQUE show distinct peaks at equally spaced ∆I intervals, which correspond to 0, 1, 2, 3, 4, 5 and 6 segments of DNA that translocate the nanopore at the same time (the last two peaks are not well dened for tD = 100 ms) [detail in Fig. 5.5.6 (a)] [33]. To compare how often multiple segments simultaneously translocate the nanopore for tD = 5 ms and tD = 100 ms, we normalize both histograms to the maximum number of samples in the rst current reduction peak, located at ∆I ≈ 0.14 nA. The peaks corresponding to 2, 3, 4, 5 and 6 folds are signicantly higher for tD = 5 ms than for tD = 100 ms. Our experimental observation that molecule translocates with multiple folds for short tD leads us to postulate that the DNA returns to the nanopore more densely packed, in a smaller coil, making it more likely for several of its segments to be captured and simultaneously translocated through the nanopore. As we increase the time between successive translocations, tD +tR , we expect that the DNA's folding will decrease. We estimate the average folding of a molecule during translocation with the mean current blockade ∆I , and calculate ∆I N>1 /∆I 1 for each recapture in a ping- pong sequence. Detail in Fig. 5.5.6 (b) shows two representative N >1 translocation signals overlaid with their N = 1 translocation signal. The molecule probed after ≈ 10 ms [detail, left] displays deep but short blockages, whereas the one detected after ≈ 330ms [detail, right] resembles its N =1 translocation signal. The former signal corresponds to a ratio ∆I N>1 /∆I 1 of approximately 2.7, whereas the latter corresponds to a ratio close to 1. Figure 5.5.6 (b) plots h∆I N>1 /∆I 1 i as a function of htD + tR i averaged over all three nanopores, where the error bars correspond to standard deviations of the mean within each 5 ms-wide bin. Average folding exhibits almost double enhancements for small tD + tR , which smoothly vanish with increasing time between two successive translocations. 76 CHAPTER 5. NON-EQUILIBRIUM DYNAMICS OF λ DNA PROBED BY THE MOLECULAR PING-PONG TECHNIQUE Figure 5.5.6: (a) 4 µs-long ∆I samples from baseline-corrected events per 0.005 nA bin, including 0.2 ms of I before and tD = 5ms (black) and after each event for tD = 100ms (gray). (b) Average folding of recaptured molecules ∆I N>1 /∆I 1 binned in 5 ms-wide bins of tD + tR . For clarity, we do not show 450 < tD + tR < 900 ms bins, which follow the trend. 77 CHAPTER 5. NON-EQUILIBRIUM DYNAMICS OF λ DNA PROBED BY THE MOLECULAR PING-PONG TECHNIQUE Similarly to Plesa et al., who studied folding of a single molecule by recapturing it hundreds of times, we nd that a large ensamble of recaptured molecules folds more for shorter times between its successive translocations. Our observation that ∆I histograms strongly depend on tD are consistent with the picture of a molecule being in a more compressed state after translocation. The continuous decrease in the average folding with increasing tD + tR suggests that molecules re-equilibrate. 5.5.4 Characterizing DNA relaxation with the measurements of average translocation speed Indications for DNA relaxation have emerged both in measurements of tR and ∆I . Here we are interested in quantifying those eects by studying average transloca- tion speed, which is inversely related to our ECD measurements. Figure 5.5.7 plots histograms corresponding to ECD measurements of subsequent recaptures, ECDN>1 (black), overlaid against their initial, N = 1 measurements, ECD1 (yellow), for all values of tD . Looking across the rst column, we observe that translocations obtained a lower ECD upon a molecule's recapture than on its rst capture by the nanopore, with a discrepancy that grew as tD decreased. Note that the histograms of N =1 measurements have roughly an order of magnitude less events than their overlaid N >1 histograms. The second column in Fig. 5.5.7 shows ECD histograms obtained at a later time in the experiment at the same nanopore, where the label t corresponds to the time when the data was collected with respect to the start of the experiment. We still observe shifts in the histograms, however, the N > 1 ECD distributions have become wider. 78 CHAPTER 5. NON-EQUILIBRIUM DYNAMICS OF λ DNA PROBED BY THE MOLECULAR PING-PONG TECHNIQUE In order to compare N > 1 ECD measurements to their corresponding N = 1 values, we build histograms of ECDN /ECD1 and show two of those, for tD = 5 ms and tD = 100 ms, in Fig. 5.5.8. The histograms are centered at 0.95 for tD = 100 ms and at 0.82 for tD = 5 ms. From Eq.5.5.1 we nd that the lower ECD of recaptured molecules reects their higher average translocation speed since the shift in ECD dis- tributions cannot be attributed to changes in ∆I . The current blockade distributions in Fig. 5.5.6 (a) show that ∆I levels coincide and are equally spaced for dierent tD . The observation that DNA translocate on average faster for short tD suggests that they initiate translocation from a more compressed conguration. In that case we ex- pect that the average translocation speed will decrease with increase in time between successive translocations, tD +tR , indicating relaxation of DNA molecules. Figure 5.5.9 plots hECDN>1 /ECD1 i as a function of htD + tR i, where error bars represent stan- dard deviations of the mean within each 5 ms-wide bin, and the measurements were averaged over all three nanopores. Molecules probed shortly after the translocation show fractional reduction from ECD1 of approximately 0.8, followed by a smooth and steady recovery towards 1 with increasing time between successive translocations. 5.5.4.1 Modeling the dependence of ECD measurements on the size of DNA coil at the moment of capture We explain the shift towards lower ECD with decreasing tD , illustrated in Fig. 5.5.8, with the reduced uidic drag on a more compact coil. The electrophoretic force, fE , pulling the DNA segments through the nanopore is opposed by viscous drag forces, fD [6, 11, 12, 13, 14, 23], whose two major contributions, fD = (ζpore + ζcoil )v , originate from the drag on the segment moving through the nanopore and the drag on the segments being pulled towards it, respectively. The average drag coecient inside 79 CHAPTER 5. NON-EQUILIBRIUM DYNAMICS OF λ DNA PROBED BY THE MOLECULAR PING-PONG TECHNIQUE Figure 5.5.7: Overlaid histograms of ECDN>1 (black) with histograms of their ECD1 (yellow) for intact λ DNA, collected on nanopore A. Histograms are ordered according to increasing tD . Start of data acquisition when we change tD is noted and labels the histograms as t. The total number of N > 1 measurements in each of the histogram is on the order of 1, 000, whereas the total number of N = 1 measurements is on the order of 100. 80 CHAPTER 5. NON-EQUILIBRIUM DYNAMICS OF λ DNA PROBED BY THE MOLECULAR PING-PONG TECHNIQUE Figure 5.5.8: Overlaid histograms of ECDN>1 /ECD1 with 0.03-wide bins for tD = 5ms (black) and tD = 100ms (gray). Detail illustrates the rst capture of the DNA by the nanopore and molecules recaptured after dierent tD . The red arrow indicates the magnitude of the mean translocation speed. the nanopore, ζpore , is constant throughout translocation [14]. The drag coecient of the segments outside the nanopore, ζcoil , exhibits non-trivial dependence on the distance across which polymer segments are pulled in towards the nanopore [12, 13]. On average, the distance across which segments are dragged into the nanopore can be taken as the radius of the DNA coil at the moment of capture which can be expressed as the radius of gyration, Rg . Even though the radius of gyration is a constant customarily used to describe the size of DNA coil in equilibrium here we use it to denote the size of a DNA coil at the moment of capture, which may arrive to the nanopore in a compressed out-of-equilibrium state. The average drag coecient is then ζcoil = αRg , where α is a constant [51, 52]. From v = L/Ttr and Eq.5.5.1 we obtain: L ECD = ∆I (ζpore + αRg ) (5.5.2) fE 81 CHAPTER 5. NON-EQUILIBRIUM DYNAMICS OF λ DNA PROBED BY THE MOLECULAR PING-PONG TECHNIQUE Figure 5.5.9: ECDN>1 measurements of recaptured molecules, normalized with their initial ECD1 , binned in 5 ms-wide bins of tD +tR and averaged over all three nanopores. Red line is a least-mean-squares t of Eq.5.5.5 to all unbinned data points for all three nanopores. For clarity, we do not show 450 < tD + tR < 900 ms bins, which follow the trend. We do not consider isolated N > 1 translocations in the ping-pong sequence with ECD larger than 2 ECD1 , presumably stuck to the nanopore (≤ 1% cases). 82 CHAPTER 5. NON-EQUILIBRIUM DYNAMICS OF λ DNA PROBED BY THE MOLECULAR PING-PONG TECHNIQUE As illustrated by the detail in Fig. 5.5.8, the DNA molecules with coil size corre- sponding to more extended congurations experience larger drag and consequently translocate slower than an initially more compact coil of lesser radius. 5.5.4.2 The model of DNA relaxation The simplest model of DNA coil re-equilibration may be obtained by treating the entire DNA coil as a single elastic spring [21, 22] displaced from equilibrium by δRg = Rgeq − Rg . By balancing the linear response of the entropic spring force of the DNA coil felastic , with the drag force, fD , acting on the coil, felastic = fD , we get: 3kB T dRg eq 2 δRg = ζ (5.5.3) (Rg ) dt We can now solve for relaxation of the DNA coil from an initially compressed state of radius Rc < Rgeq : Rg (t) = Rgeq − (Rgeq − Rc )e−t/τ (5.5.4) where τ = ζ(Rgeq )2 /3kB T sets the longest relaxation time scale, on which the entire DNA coil re-equilibrates. Such a model easily includes or ignores hydrodynamic interactions by re-dening ζ. The Zimm model includes hydrodynamic interactions by eectively treating the DNA coil as an impermeable sphere giving rise to ζ = 6πηRgeq [21, 22], where η is solvent viscosity. In the Rouse model, the DNA coil is modeled as a set of N beads connected by springs and assumed to be freely-draining. Here ζ = ζbead N , which linearly scales with DNA's total length. 83 CHAPTER 5. NON-EQUILIBRIUM DYNAMICS OF λ DNA PROBED BY THE MOLECULAR PING-PONG TECHNIQUE 5.5.4.3 ECD measurements reveal the relaxation time of DNA We now obtain an explicit dependence of our ECD measurements on tD + tR by com- bining Eq. (5.5.2) and Eq. (5.5.4). In order to t our data in Fig.5.5.9, we normalize  Eq. (5.5.2) with equilibrium value ECDeq = ∆IL ζpore + αRgeq /fE . By assuming that ECD1 = ECDeq , we obtain: ECDN>1 (t) = 1 − Ae−t/τ (5.5.5) ECD1   where A = α Rgeq − Rc / ζpore + αRgeq and the re-equilibration time τ are our t parameters. The solid line in Fig. 5.5.9 plots the re-equilibration curve of λ DNA molecules predicted by Eq. (5.5.5) with A = 0.175±0.002 and τ = 106±5 ms obtained from the least mean squares t to all unbinned translocation data collected by three dierent nanopores. We nd that the a simple model of DNA relaxation shows excellent agreement with our data. The value of relaxation time τ is comparable to values that can be found in the literature [46, 48]. Furthermore, from the t parameter A we obtain the ratio of average drag coecients for the compressed and equilibrated coil, (αRc + ζpore )/(αRgeq + ζpore ) = 0.825. 5.6 Discussion We found that the recapture time distributions predicted by the drift and diusion of a structureless Brownian particle, shown in Fig. 5.5.2, do not t our measurements for short tD . At longer tD , the model converges to recapture data histograms, but still with a non-negligible discrepancy in the mean values of approximately 20%. We 84 CHAPTER 5. NON-EQUILIBRIUM DYNAMICS OF λ DNA PROBED BY THE MOLECULAR PING-PONG TECHNIQUE explain this nding as being indicative of DNA expansion, which ultimately leads to hastened recaptures. Additionally, the expansion of the DNA during tD = 5 ms happens in the steep gradient of electric eld; Fig. 5.5.3 (b) shows the distribution of the center of mass of the DNA that migrated away from the nanopore for tD = 5 ms and Fig. 5.5.3 (a) shows the prole of the electric eld in the vicinity of a nanopore. Due to the DNA expansion, the trailing segments will end up being under the inuence of stronger electric eld after the reversal of the driving potential, which may further contribute to hastened DNA recaptures. However, a non-negligible gradient of electric eld in the vicinity of the nanopore may also aid the recapture of the DNA by stretching its segments towards the nanopore. Furthermore, the diusion coecient for the motion of center of mass away and towards the nanopore may no longer be the same, since DNA expands. In our computation of recapture rates we did not take these eects into account. The monotonic increase in ECDN>1 measurements shown in Fig. 5.5.9 is consistent with the picture of the DNA coil freely expanding from a compact out-of-equilibrium state. Our intuitive picture of DNA relaxation portrays isotropic expansion of DNA coil from a spherically symmetric compressed conguration. However, since the speed at which the DNA segments are being fed through the nanopore during translocation is much larger than the speed at which the translocated segments are migrating away from it, DNA segments coming out of the nanopore are forced to pile on top of each other and may start spreading laterally away in the plane parallel to the nanopore due to excluded volume interactions. Such a pancake-like initial conguration [53] may ultimately lead to hastened recaptures since it would provide more lingering segments available for capture. This arrangement would also lead to translocations with multiple folds and on average faster translocations as the DNA is captured from 85 CHAPTER 5. NON-EQUILIBRIUM DYNAMICS OF λ DNA PROBED BY THE MOLECULAR PING-PONG TECHNIQUE a distance closer relative to the nanopore. Simulations may oer novel insight into these conformations and estimate their inuence to our experiment [54]. In our translocation model we assumed ζpore = const. and ζcoil = αRg . However, recent theoretical models and simulations show that the tension force propagating along the backbone of the polymer when it is being pulled into the nanopore may result in DNA forming a trumpet-like shape or a stem-ower-like shape [15, 16, 17, 27]. This may alter our estimate of the average drag outside the nanopore. 5.7 Conclusion In our experiments we implemented the molecular ping-pong technique to access dynamics of polymer relaxation outside the nanopore and measure the time τ in which the whole coil re-equilibrates after translocation. We have shown that the fast translocation process drives the DNA into a compressed state and the subsequent recapture quanties its re-equilibration via the ionic current blockade signal. We found three pieces of evidence suggesting that rapid translocation of the DNA through the nanopore drives a DNA into a compact, out-of-equilibrium congura- tion, from which it expands on the scale of the longest relaxation time. For short tD we found that molecules in out-of-equilibrium states are recaptured faster than Brownian particles of same mobility and diusion constant, that they translocate on average faster and with multiple folds. By increasing time between successive translocations we observe reduction in folding and decrease in average translocation speed, compatible with molecules that are approaching their equilibrium congura- tions. Experimental evidence points to a simple relaxation model that provides an exceptionally good t to our ECD data. 86 Appendix 5.A Description of experimental limitations of the ping-pong technique In the ping-pong experiment, we have allowed for DNA to be recaptured up to 26 times, waiting for each recapture for 5tD . However, we found that our ping-pong sequences would sometime terminate before the maximal number of recaptures was achieved. The analysis revealed a few reasons for that: (1) our Labview software failed to detect a DNA translocation (2) sometimes a second translocation event was detected before the voltage reversal and (3) sometimes the DNA molecule needed more that 5tD to return to the nanopore. The root cause for the failure our software to detect a DNA translocation is the capacitative charging of the membrane in which the nanopore is embedded. Molecules that appeared on the slope of I sometimes did not meet the positive or negative threshold in current, Ith , for event detection. 87 CHAPTER 5. NON-EQUILIBRIUM DYNAMICS OF λ DNA PROBED BY THE MOLECULAR PING-PONG TECHNIQUE Figure 5.A.1: Current versus time trace of ve consecutive translocations of DNA through the nanopore for tD = 5 ms. N = 2 and N =4 DNA translocations do not trigger voltage reversal; consequently we measure unusually long tD for N =3 and N =5 translocations. 88 CHAPTER 5. NON-EQUILIBRIUM DYNAMICS OF λ DNA PROBED BY THE MOLECULAR PING-PONG TECHNIQUE 5.A.1 Distributions of measured delay times Figure 5.A.1 displays a ping-pong sequence of ve consecutive DNA translocations, which clearly shows that time following N = 2 and N = 4 translocation events is much longer than tD = 5 ms, which we chose for this experiment. When the molecule was recaptured quickly after the voltage reversal, the translocation signals, such as N =2 and N = 4, appeared on the steep slope of I and failed to cross Ith for the duration of 0.3 ms in order to trigger the voltage reversal. Our Labview software is set to restore positive baseline current by reversing the voltage back to its positive polarity after 5tD . The ping-pong sequence can still continue if the DNA reappears and triggers another voltage reversal, such as the N =3 event in Fig. 5.A.1. As a consequence, the N =3 and N = 5, obtained unusually long tD . Figure 5.A.2 plots histograms of measured delay times, binned in 0.1 tD -wide bins, for each of pre-set tD = 5, 10, 20, 30, 50, 100 and 150 ms. All histograms show that majority of events is centered around the pre-set tD . However, we also observe a smaller number of events forming a peak centered close to 5tD , which grew as tD decreased. For tD = 5 and 10 ms we also observe a non-negligible contribution from events with shorter delay times. These are probably measured when a higher order fold along the DNA molecule, appearing on slope of I, triggers the voltage reversal and the counting of tD starts before the molecule leaves the nanopore. 5.A.2 Recaptures within tR-resolution Figure 5.A.3 (a) shows an event recaptured 0.2 ms after the voltage reversal, appearing on the steepest slope of I. For such events we are unable to discern their beginning since I and the current blockades, ∆I , have comparable slopes. Furthermore, the 89 CHAPTER 5. NON-EQUILIBRIUM DYNAMICS OF λ DNA PROBED BY THE MOLECULAR PING-PONG TECHNIQUE Figure 5.A.2: Histograms of measured delay times for pre-set tD in the range of 5−150 ms. Translocation events mostly have measured delay times within ∼ 10% of the pre-set tD . We measured either very long or a short delay times for a large fraction of events when we chose tD = 5 ms. This eect decreases with increasing tD . 90 CHAPTER 5. NON-EQUILIBRIUM DYNAMICS OF λ DNA PROBED BY THE MOLECULAR PING-PONG TECHNIQUE polynomial t does not adequately approximate the slope of I within rst 0.2 ms after the voltage reversal. Figure 5.A.3 (b) displays the same event after baseline t subtraction, showing that we could not restore the beginning of the event. For such events measurements of ECD and ∆I are not trustworthy. Our inability to precisely resolve the start time of these events sets the recapture time resolution limit to tR = 0.19 ms. We only account for these events in the rst bins of tR histograms. Figure 5.A.3: Current trace of a translocation event occurring within 0.2 ms after the voltage reversal. (a) Event superimposed on the steepest slope of I. (b) Event after baseline t subtraction. We can not resolve the beginning of the event. 5.A.3 Double-trouble Occasionally, a second molecule can be captured during tD , before the voltage is reversed. Figure 5.A.4 shows that after 9 consecutive translocations of a molecule through the nanopore, another molecule got captured during the 50 ms-long delay time. After the voltage reversal both molecules got recaptured [detail, bottom left], and then only one continues the ping-pong sequence [detail, bottom right]. In this 91 CHAPTER 5. NON-EQUILIBRIUM DYNAMICS OF λ DNA PROBED BY THE MOLECULAR PING-PONG TECHNIQUE specic case, the molecule that continues the ping-pong sequence has ECD that sums up to the ECD of both separate molecules. Since we are unable to determine with precision which molecule continues the ping-pong sequence, we do not consider any of subsequent translocations within such sequences. Figure 5.A.4: Current versus time trace of a portion of a ping-pong sequence showing capture of a second molecule between N = 9 translocation event and the voltage reversal. Bottom left: A detail showing both molecules getting recaptured. Bottom right: A detail showing a single translocation event with large ECD. 92 CHAPTER 5. NON-EQUILIBRIUM DYNAMICS OF λ DNA PROBED BY THE MOLECULAR PING-PONG TECHNIQUE 5.A.4 Molecules stuck to the nanopore during translocation We measured isolated translocation events with unusually large ECD within a ping- pong sequence. Figure 5.A.5 shows one such sequence with a single translocation event, the N = 4, which obtained an unusually large ECD. We assume that the DNA interacted with the nanopore during the N = 4 translocation. We use ECD to measure the change in non-equilibrium DNA coil size. Events that get stuck to the nanopore cannot provide such an information. In our analysis we do not consider recaptures of the molecule with ECD two times larger than measured during its rst translocation through the nanopore. This happens in less than 1% of all cases. Figure 5.A.5: Current versus time trace a ping-pong sequence of six successive translo- cations showing an unusually long translocation signal during the N =4 transloca- tion, probably due to a molecule that got stuck to the nanopore. 93 CHAPTER 5. NON-EQUILIBRIUM DYNAMICS OF λ DNA PROBED BY THE MOLECULAR PING-PONG TECHNIQUE 5.A.5 Distributions of DNA recaptures In our experiment, we allowed DNA to be recaptured within 5tD after each voltage reversal, for up to 26 times. We noticed that the number of successful recaptures of a single molecule in a ping-pong sequence depends on tD . Figure 5.A.6: Histograms show the number of times each molecule got recaptured (where 0 corresponds to no recaptures and our Labview software allows up to 26 recaptures) for pre-set tD in the range of 5−150 ms. For each histogram, the number of molecules per recapture bin is normalized by the total number of molecules, m, giving a fraction of molecules that got recaptured up to 26 times. In order to verify that observation, in Fig. 5.A.6 we plot histograms for all tD values used in our experiment, in which we bin molecules probed by nanopore A according to the total number of their successful recaptures (divided by the total 94 CHAPTER 5. NON-EQUILIBRIUM DYNAMICS OF λ DNA PROBED BY THE MOLECULAR PING-PONG TECHNIQUE number of molecules, m). We nd that the probability of not recapturing the molecule at all steadily grew with increasing delay time, leaving us with ∼ 15% chance that we immediately lose a molecule after its rst translocation for tD = 150 ms. Furthermore, for very long delay times a very low number of molecules achieved maximal number of recaptures. Our histograms show that more than 40% of the total DNA molecules probed for tD = 10 ms and tD = 30 ms got recaptured 26 times, whereas that happened in only ∼ 10% cases for tD = 100 ms, and in only a few percent cases for tD = 150 ms. Figure 5.A.7: Probability of recapturing molecules for the maximum of 26 times. Data is collected on three dierent nanopores, for the choice of tD between 2 and 150 ms. We observe greatest recapture probabilities for tD in the range 5-30 ms. In Fig. 5.A.7 we plot the fraction of molecules that got recaptured 26 times, for all values of tD , collected on three dierent nanopores. We observe that the experiment is most likely to achieve maximum number of recaptures for tD between 5 and 30 ms. Outside of this interval of delay times the probabilities drop to less than 20%. 95 CHAPTER 5. NON-EQUILIBRIUM DYNAMICS OF λ DNA PROBED BY THE MOLECULAR PING-PONG TECHNIQUE Figure 5.A.8: Probability of recapturing intact λ DNA, calculated by using Eq. 5.A.1. We observe largest probabilities for a choice of tD between 5 and 30 ms. To quantify the eciency of DNA recaptures, we dene recapture probability as: P26 i × b(i) i=0 P = P25 (5.A.1) i=0 ((i + 1) × b(i)) + 26 × b(26) where b(i) denotes the number of molecules that got recaptured i times. In the for- mula, we divide the average number of recaptures by the average number of attempts. Figure 5.A.8 plots recapture probabilities, for all three nanopores, as a function of de- lay time. The recapture probability is largest for tD between 5 and 30ms, for all three nanopores. For very short tD , lower recapture probability can be attributed to technical challenges. The ping-pong sequence was often interrupted if a molecule appeared on the steep slope of I and failed to trigger voltage reversal. This would happen only when positive voltage bias was applied, as explained in Section 5.A.1. For tD > 30 ms recaptures are again less probable. In those cases molecules may need 96 CHAPTER 5. NON-EQUILIBRIUM DYNAMICS OF λ DNA PROBED BY THE MOLECULAR PING-PONG TECHNIQUE more than 5tD to be recaptured. Furthermore, a second molecule may get recaptured before voltage reversal, in which case we discarded all subsequent passes. This was more likely to happen for long delay times, occurring in 10% of cases for tD = 100 ms. This exclusion procedure slightly inuences recapture probabilities. 5.B Analysis of translocation signals Here we describe the analysis of translocation events with a custom MatLab software. Figure 5.B.1 shows two typical time recordings of ionic current through the nanopore, collected for over 45 seconds of the experiment for tD = 5 ms and tD = 150 ms, consist- ing of many ping-pong series separated by at least 1-second-long intervals of positive baseline current. For tD = 5 ms, a single ping-pong series is a region of densely packed current spikes alternating in polarity between −200 and 200 nA [Fig. 5.B.1 a)]. The width of these regions indicates the number of recaptures within a ping-pong se- quence. For tD = 150 ms, within one ping-pong sequence we clearly observe alter- nating polarity of I, accompanied by capacitative spikes [Fig. 5.B.1 b)]. Within each sequence translocations can be observed as transient blockades, indicating presence of a molecule being bounced back and forth multiple times, such as shown in Fig. 5.B.2. In our MatLab analysis we rst group the translocation events that belong to multiple recaptures of a single molecule, then we subtract the t to the I between each two voltage reversals to obtain DNA translocation events on the attened baseline, and nally we analyze the current blockades, by measuring their start and end times, ECD, current blockade levels, and delay and recapture times. 97 CHAPTER 5. NON-EQUILIBRIUM DYNAMICS OF λ DNA PROBED BY THE MOLECULAR PING-PONG TECHNIQUE Figure 5.B.1: Time recoding of 45-second-long current traces. a) Ping-pong sequences correspond to vertical strips of current alternating between −200 and 200 nA, collected for tD = 5 ms. b) Ping-pong sequences recorded for tD = 150 ms show consecutive changes in polarity of I, accompanied by large spiking. In both cases, sequences are separated by at least 1-second-long positive current. 98 CHAPTER 5. NON-EQUILIBRIUM DYNAMICS OF λ DNA PROBED BY THE MOLECULAR PING-PONG TECHNIQUE 5.B.1 Detection of ping-pong sequences We perform all of our analysis on the absolute value of I. The analysis software starts by searching the time recording of the ionic current for all capacitative spikes, and records the time position of these current spikes as the absolute value of I crosses the threshold of 180 nA. We group current spikes into a ping-pong sequence of a single molecule when the time dierence between two successive spikes is less than 900 ms. We can use this criterion since the longest time between two voltage reversals within the ping-pong sequence is set to 5tD , which in our experiment is never greater than 750 ms. After we have grouped the current spikes into ping-pong sequences, we dene the 3tD -long current trace before the rst voltage reversal in the group as the beginning of a ping-pong sequence and the 6tD -long current trace after the last voltage reversal in the group as the end of a ping-pong sequence. One such sequence is shown in Fig. 5.B.2, which is the rst visual output of our software. 5.B.2 Fitting the slope of I after the voltage reversal 1 We implemented the BackCor subroutine in our MatLab analysis to t the decaying slope of I. The BackCor subroutine allows us to choose the type of the polynomial, the polynomial order and the tolerance that we want to use to t I. For each tD we chose a symmetric truncated polynomial of 50th order and 0.03 tolerance to t a single baseline with no event present. We tted the baseline starting from 0.1 ms after voltage reversal (when current spike crosses 180 nA-threshold) until either I reached steady state or for 5tD , whichever came rst. For nanopore A we measured that the baseline reaches its steady-state after 100 ms (for nanopore B that happens after 30 ms 1 freely available for download at MatLab Central, http://www.mathworks.com/matlabcentral/ fileexchange/27429-background-correction/content/backcor.m 99 CHAPTER 5. NON-EQUILIBRIUM DYNAMICS OF λ DNA PROBED BY THE MOLECULAR PING-PONG TECHNIQUE Figure 5.B.2: First visual output of our analysis software is a ping-pong sequence showing multiple recaptures of a single DNA molecule. The sequence starts 3tD before the rst voltage reversal and ends 6tD after the last voltage reversal. 100 CHAPTER 5. NON-EQUILIBRIUM DYNAMICS OF λ DNA PROBED BY THE MOLECULAR PING-PONG TECHNIQUE and for nanopore C after 50 ms). Figure 5.B.3 (a) shows the result of a polynomial t to the baseline for tD = 10 ms and Fig. 5.B.3 (b) displays the baseline after subtraction of the polynomial t. We used Fig. 5.B.3 (b) to estimate the goodness of the chosen polynomial t, where a good t is to approximate the rst few miliseconds after the voltage reversal well, shown in the detail. However, none of the BackCor parameters allowed a good t for the very rst 0.2 ms after the voltage reversal. Figure 5.B.3: Fitting the slope of I after the voltage reversal. (a) The BackCor subroutine ts the current trace recorded after the voltage reversal with no event present. (b) Subtraction of the polynomial t from the baseline tests how well chosen polynomial approximates the decay of I. 101 CHAPTER 5. NON-EQUILIBRIUM DYNAMICS OF λ DNA PROBED BY THE MOLECULAR PING-PONG TECHNIQUE 5.B.3 Translocation event detection Figure 5.B.4 (a) shows the visual output of our MatLab software, which displays the current trace between two spikes previously detected within each ping-pong sequence. The red solid line shows a saved baseline t function laid over the current trace by setting the tail of both the t and the baseline to zero. The current trace was set to zero by subtracting the mean of the last 0.2 ms before the voltage reversal and the polynomial t was set to zero by subtracting the mean of 0.2 ms at the end of its length (it either ts the full 5tD of the baseline or until it reaches steady-state). We then subtract the t from the absolute value of I between two successive voltage reversals within a ping-pong sequence. Figure 5.B.4: Visual output of analysis software. (a) Saved baseline t function is laid over the current trace recorded between two successive voltage reversals. (b) Baseline t function in the vicinity of current blockade event on the slope of I. (c) Current blockade event (shown in blue) after subtracting the baseline t function and subtracting the average of 0.2 ms-long baseline after the event (shown in green). Note that 250 time steps correspond to 1 ms. 102 CHAPTER 5. NON-EQUILIBRIUM DYNAMICS OF λ DNA PROBED BY THE MOLECULAR PING-PONG TECHNIQUE After attening the baseline by subtracting the BackCor polynomial t, we search for the event by looking for current blockade crossing the threshold set at 7 RMS (this corresponds to ∼ 0.5∆I ) below baseline. The RMS (Root-Mean-Square) of the baseline is determined once for each nanopore by calculating the square root of the average quadratic deviation from zero of a 3 ms-long piece of at baseline. All three nanopores had comparable RMS ' 0.01 nA. After we coarsely nd the beginning and the end of an event we subtract the average of a 0.2 ms-long baseline after each event (the baseline before was often too close to the voltage reversal), shown in green in Fig. 5.B.4 (c), to set baseline to zero. We then measure the start and the end time of an event when the current blockade crosses 2 RMS ≈ 0.15∆I threshold below baseline. Figures 5.B.4 (b) and (c) are a visual output of the software, showing the detected event before and after baseline correction. Figure 5.B.4 (c) displays the event with at-bottomed quantized current blockade steps, ∆I , painted in blue from its start to the end. 5.C Numerical evaluation of the drift-diusion equa- tion We model the recapture time distributions by using Smoluchovski equation. Since the electric eld decays as 1/r2 away from the nanopore, the problem is spherically symmetric (actually, hemispherically, as we have the nanopore chip wall on one side) and the Smoluchowski equation can be written as   ∂c(r, t) 1 ∂ 2 µ |I| ∂c(r, t) = 2 r ∓ c(r, t) + D (5.C.1) ∂t r ∂r 2πσr2 ∂r 103 CHAPTER 5. NON-EQUILIBRIUM DYNAMICS OF λ DNA PROBED BY THE MOLECULAR PING-PONG TECHNIQUE where c(r, t) is the volume concentration of DNA, µ is the electrophoretic mobility, σ conductivity of solution and D diusion constant, which are given in the main text. Physically, 2πr2 c(r, t)dr corresponds to the probability of nding the DNA at time t in a hemispherical shell at radial distance between r and r + dr away from the nanopore. The rst term is due to the electrophoretic drift in an electric eld E = ∓|I|/(2πσr2 ), where the minus sign describes the motion of DNA away from the nanopore, and positive sign to DNA moving towards the nanopore. The second term is due to random diusive forces on the DNA. The two competing eects, drift and diusion, dominate in dierent regimes. This can be qualitatively evaluated by comparing the velocity of DNA due to electropohoretic drift, v = µI/2πr2 σ , and average radial velocity of diusion, vD = D/r [8]. Equating the two velocities we arrive at a characteristic distance beyond which diusion starts dominating over drift, L = µI/2πσD. In our experimental setup L = 9 µm. Our partial dierential equation solver, pdepe, which comes as a standard package in MatLab, works only with dimensionless variables, so we transform Eq. 5.C.1 to   ∂c(x, s) 1 ∂ 2 ∂c(x, s) = 2 ∓c(x, s) + x (5.C.2) ∂s x ∂x ∂x where we dened s = t/τ and x = r/L and we used τ = L2 /D. τ corresponds to the time in which the molecule would diuse a distance L away from the nanopore, setting the maximal time scale in our ping-pong cycle. For the 48-kbp long λ DNA we obtain τ = 138 s, which is many orders of magnitude larger than delay and recapture times we consider. 104 CHAPTER 5. NON-EQUILIBRIUM DYNAMICS OF λ DNA PROBED BY THE MOLECULAR PING-PONG TECHNIQUE After the translocation the DNA is located a distance r0 away from the nanopore. Our initial condition is therefore a delta function, δ(r − r0 ) c(r, t = 0) = (5.C.3) 2πr02 which can be numerically approximated with a very narrow Gaussian centered at r0 . We chose r0 = 100 nm, and the width of the Gaussian σ = 1 nm comparable to the spacing in the numerical grid on which we solve Eq. 5.C.2. We tested that our result does not depend on the choice of σ. For the motion away from the nanopore we imposed the no-ux boundary conditions at the position of the nanopore and at innity (rmax ): ∂c(r, t) ∂c(r, t) = =0 (5.C.4) ∂r r=0 ∂r r=rmax where rmax = 6 µm is the largest distance for which we numerically evaluate the solution. We made sure that the chosen value of rmax is large enough so that the solutions do not strongly depend on our choice. By using Eq. 5.C.2 we numerically evolve the initial condition, Eq. 5.C.3, for motion of DNA away from the nanopore. The solution obtained at time t = tD is an initial condition for the molecule's return. The boundary condition at the nanopore now needs to change to absorbing, c(rc , t) = 0 (5.C.5) while the boundary condition at r = rmax stays the same. rc is the minimal distance the DNA has to approach the nanopore in order to be captured by the nanopore and translocate through. In our calculations, we chose it to be slightly smaller than the distance from which molecule started moving away, rc = 90 nm. In our computations, 105 CHAPTER 5. NON-EQUILIBRIUM DYNAMICS OF λ DNA PROBED BY THE MOLECULAR PING-PONG TECHNIQUE we found that the prole of our solutions did not strongly depend on the choice of rc . Finally, we can construct the rate at which the DNA is recaptured by the nanopore, giving us probability of observing translocation at a given value of tR . We rst integrate the volume concentration over the volume of the uidic compartment to get total probability of nding the molecule, and then compute the capture rate C(t), which is negative change in probability of observing a molecule between time t and t + dt, anywhere inside the uidic compartment: ˆ d C(t) = − 2πr2 c(r, t)dr (5.C.6) dt The predictions of our model, normalized to the maximum rate, are plotted in Fig. 5.5.2 and Fig. 5.5.3. 106 Chapter 6 Outlook 107 CHAPTER 6. OUTLOOK In this thesis we experimentally accessed many interesting aspects of polymer physics, which are relevant at scales accessible to a nanopore. These small detectors, designed to thread nanometer-thin charged polymers, have proved their usability through mapping of DNA lengths and measuring forces acting on DNA when pulled through tight constrictions. Even though these measurements seem to relate to the instantaneous properties of a part of the molecule that is being threaded through, in our experiments we managed to discover that the nanopore can also tell us more about what the molecule was doing before being captured by the nanopore. This is in itself a remarkable nding, as it opens a whole array of studies on single molecules unperturbed from their natural habitat, as they are freely suspended in solution. In this work we experimentally veried two important concepts, the congurational entropy that is sucient to account for the experimentally observed multiplicity of congurations at the capture by the nanopore, and dynamic relaxation of the DNA as being studied by repeated translocation through a nanopore. The ping-pong technique oers a novel approach to study compression and free expansion of DNA molecules, by the means of extremely high electric elds and a few milisecond time resolution, without the need for DNA connement or chemical alterations, such is common in nanochannels [46, 55, 56, 57]. This technique allowed us to measure the re-equilibration time of λ DNA at high salt concentration. By lowering the salt concentration we can also probe the inuence of excluded volume interactions on the compression and ensuing expansion of DNA molecules. We expect that after translocation the DNA segments will form a less compressed coil due to excluded volume interactions. At lower salt concentrations, the electrostatic screening is less eective, leading to greater electrostatic repulsion between two DNA segments and therefore to the eective increase of excluded volume around the DNA strand. 108 CHAPTER 6. OUTLOOK The eective diameter of the DNA is shown to increase from ∼ 2.7 nm at 1 M KCl to ∼ 11 nm at 20 mM [22]. In nanopore experiments we would be able to detect DNA translocations at ionic concentrations as low as 50 mM [31]. By comparing relaxation times and relative compression of λ DNA at high and low salt concentrations, we can infer about the importance of excluded volume interactions on the dynamics of charged polyelectrolytes in solution. Immediately after translocation, we expect λ DNA will be located at a distance of around 100 nm away from the nanopore. In the presence of the electric eld, the DNA drifts another 500 nm within rst 5 ms after translocation (cf. Fig. 5.5.3 (b)). Due to the presence of the nanopore wall, the diusion also on average pushes the DNA away, but at a much slower rate; in 5 ms the DNA would diuse only another 100 nm away from the nanopore. We expect the diusion to become less directed at later times. Therefore, turning o the electric eld after the translocation in a ping- pause-pong experiment will let the DNA coil stay reasonably close to the nanopore for extended periods of time. By comparing the recapture time distributions in ping- pong and ping-pause-pong experiments, we can quantify the eects of the electric eld on the DNA coil in solution. Furthermore, we expect this situation will increase recapture probabilities and allow us to probe larger molecules that relax on much longer timescales. Moreover, the ping-pong technique allows us to probe individual DNA molecules multiple times and thereby can improve the accuracy of length maps. Our ongoing research in this direction suggests that subsequent measurements of the DNA length are weakly correlated. In another experiment we can use a ladder of dierent DNA lengths, and by averaging over subsequent measurements in a ping-pong sequence we expect that the dispersion in measured contour lengths will decrease. The possibility 109 CHAPTER 6. OUTLOOK of obtaining a large number of independent measurements on the same DNA molecule can help guide the development of nanopore-based DNA sequencing. Furthermore, there are many more uses and applications of nanopores, and instead of listing those that have been done or put forward by other groups, here we outline some possible future directions that our lab can explore with current experimental methods and techniques. Building on our previous work [58], we can fabricate devices with a gated nanochannel guiding the DNA into a nanopore. These have an advantage over traditional nanopores as they would iron out the coiled DNA on its arrival to the nanopore through a long slim channel, with a possibility of controlling the capture rate and speed of translocations by gating the entrance to the nanopore. We can modify already described fabrication procedure to deposit a layer of Cr on top of the nanopore in the shape of an electrode, use FIB to make a ∼ 100 nm opening and etch through the Cr to get to the minimembrane. After coating with a thin layer of aluminum oxide we are left with a nanopore in a minimembrane and a long nanochannel leading to the nanopore. Another idea is to build structures on top of a nanopore, to trap and contain the DNA molecules in the vicinity of the nanopore after their translocation [59]. For that purpose we can pierce a mezzo-pore in a 400 nm-thick SiN membrane and laterally etch silicon-dioxide [Fig. 3.1.1 (i)] to form a cylindrical chamber of about 1 µm in diameter. One could use this chamber to do single molecule chemistry, such as snipping molecules with the use of restriction enzymes, and then reverse the voltage to recapture and size each of the fragments that remain after the chemical reaction. 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