Monte Carlo implementation of supercoiled double-stranded DNA

Metropolis Monte Carlo simulation is used to investigate the elasticity of torsionally stressed double-stranded DNA, in which twist and supercoiling are incorporated as a natural result of base-stacking interaction and backbone bending constrained by hydrogen bonds formed between DNA complementary nucleotide bases. Three evident regimes are found in extension versus torsion and/or force versus extension plots: a low-force regime in which over- and underwound molecules behave similarly under stretching; an intermediate-force regime in which chirality appears for negatively and positively supercoiled DNA and extension of underwound molecule is insensitive to the supercoiling degree of the polymer; and a large-force regime in which plectonemic DNA is fully converted to extended DNA and supercoiled DNA behaves quite like a torsionless molecule. The striking coincidence between theoretic calculations and recent experimental measurement of torsionally stretched DNA [Strick et al., Science {\bf 271}, 1835 (1996), Biophys. J. {\bf 74}, 2016 (1998)] strongly suggests that the interplay between base-stacking interaction and permanent hydrogen-bond constraint takes an important role in understanding the novel properties of elasticity of supercoiled DNA polymer.Comment: 21 pages, 6 PS figures. To appear at Biophys.

Knot Invariants from Four-Dimensional Gauge Theory

It has been argued based on electric-magnetic duality and other ingredients that the Jones polynomial of a knot in three dimensions can be computed by counting the solutions of certain gauge theory equations in four dimensions. Here, we attempt to verify this directly by analyzing the equations and counting their solutions, without reference to any quantum dualities. After suitably perturbing the equations to make their behavior more generic, we are able to get a fairly clear understanding of how the Jones polynomial emerges. The main ingredient in the argument is a link between the four-dimensional gauge theory equations in question and conformal blocks for degenerate representations of the Virasoro algebra in two dimensions. Along the way we get a better understanding of how our subject is related to a variety of new and old topics in mathematical physics, ranging from the Bethe ansatz for the Gaudin spin chain to the $M$-theory description of BPS monopoles and the relation between Chern-Simons gauge theory and Virasoro conformal blocks.Comment: 117 page

Single DNA conformations and biological function

From a nanoscience perspective, cellular processes and their reduced in vitro imitations provide extraordinary examples for highly robust few or single molecule reaction pathways. A prime example are biochemical reactions involving DNA molecules, and the coupling of these reactions to the physical conformations of DNA. In this review, we summarise recent results on the following phenomena: We investigate the biophysical properties of DNA-looping and the equilibrium configurations of DNA-knots, whose relevance to biological processes are increasingly appreciated. We discuss how random DNA-looping may be related to the efficiency of the target search process of proteins for their specific binding site on the DNA molecule. And we dwell on the spontaneous formation of intermittent DNA nanobubbles and their importance for biological processes, such as transcription initiation. The physical properties of DNA may indeed turn out to be particularly suitable for the use of DNA in nanosensing applications.Comment: 53 pages, 45 figures. Slightly revised version of a review article, that is going to appear in the J. Comput. Theoret. Nanoscience; some typos correcte

Experimental data reduction for hyperelasticity

[EN] WYPiWYG hyperelasticity is a data-driven, model-free computational procedure for finite element analysis of soft materials. The procedure does not assume the shape of the stored energy function and does not employ material parameters, predicting accurately any smooth prescribed behavior from a complete set of experimental tests. However, fuzzy experimental data may yield useless highly oscillatory, unstable stored energy functions, and classical curvature smoothing frequently gives unsatisfactory results. Aside, the possibility of having experimental data from different specimens for the same test was not considered in previous procedures. In this work we present a novel technique based on spline regression and smoothing penalization using stability conditions. In general, this procedure reduces noisy experimental data or data from multiple specimens for ulterior determination of the stored energy. The procedure only needs the solution of a linear system of equations. Instead of classical curvature-based smoothing, we employ a novel stability-based smoothing, determining for each branch of the uniaxial stress-strain curve the most restrictive stability condition during uniaxial and equibiaxial tests. The resulting stored energy functions are smooth and stable. The procedure has little sensitivity to the number of spline segments or to the choice of the penalization parameter, which are computed automatically.Partial financial support for this work has been given by grant DPI2015-69801-R from the Direccion General de Proyectos de Investigacion of the Ministerio de Economia y Competitividad of SpainLatorre, M.; Montáns, FJ. (2020). Experimental data reduction for hyperelasticity. Computers & Structures. 232:1-16. https://doi.org/10.1016/j.compstruc.2018.02.01111623

Development and Application of Quantum Chemical Methods for the Description of Molecules Under Mechanical Stress

In mechanochemistry, forces are used to initiate chemical reactions. Although mechanochemical reactions have been conducted for millennia, a fundamental understanding of the relevant processes at the molecular level is still unavailable. Nevertheless, mechanochemistry is a lively field with an extraordinarily wide range of applications. Several approaches to apply forces to single molecules are commonly used today. One such approach is Single-Molecule Force Spectroscopy, where forces are transmitted from the cantilever of an Atomic Force Microscope to a macromolecule that is anchored to a glass surface. Moving the cantilever away from the surface exerts a pulling force on the molecule. In another class of mechanochemical experiments, ultrasound baths are used to rupture polymers mechanically or to transduce forces to a mechanically susceptible moiety. This capability of ultrasound baths is due to the collapse of cavitational bubbles in the liquids, which generates tensile forces. Furthermore, ball milling or grinding techniques can be used to crush solids, thereby applying forces to molecules. This procedure can be applied to make and break covalent bonds. Due to the lack of solvent, mechanochemical synthesis in a ball mill shows enormous potential as a sustainable and environment-friendly alternative to thermochemistry. Despite this rich body and long history of experimental mechanochemical procedures, the underlying processes are not well understood at the molecular level. However, such a comprehension is desperately needed for the optimization of mechanochemical syntheses. The use of quantum chemical methods to describe mechanochemical processes, which is called quantum mechanochemistry, has proven to be of tremendous value in understanding mechanochemistry at its most fundamental level. Quantum chemical methods for the description of molecules under external forces afford predictions on force-induced changes in molecular geometry, reactivity and spectroscopic properties. Moreover, force analysis tools are available that can be used to identify the mechanically relevant degrees of freedom in a molecule or its force-bearing scaffold, thereby rationalizing mechanochemical reactivity. During my PhD work, I have developed the JEDI (Judgement of Energy DIstribution) analysis, which is a quantum chemical force analysis tool for the distribution of stress energy in a mechanically deformed molecule. Based on the harmonic approximation, an energy is calculated for each bond, bending and torsion in a molecule, thus allowing the identification of the mechanically most strained regions in a molecule as well as the rationalization of mechanochemical processes. When a molecule is stretched, some internal modes store more energy than others. This leads to particularly large displacements of certain modes and to the preconditioning of selected bonds for rupture. Using the JEDI analysis I investigated the mechanochemical properties of polymer strands that are tangled into knots. In analogy to ropes, polymer strands are weakened by the ubiquitous overhand knot by approximately 50% and the point of bond rupture is located at the “entry” or “exit” of the knot. The JEDI analysis revealed the reason for this behavior. Upon stretching, most stress energy is stored in the torsions of the curved part of the knot and only a remarkably small amount of energy is used to stretch the bonds that ultimately break. This observation leads to the physical picture that the knot “chokes off” the chain in its immediate vicinity. In this process, the torsions act as work funnels that effectively localize the mechanical energy in the knot, thus preconditioning the covalent bonds at its entry and exit for bond rupture. Besides the description of mechanical deformation in the ground state, the JEDI analysis can be used in the electronically excited state to quantify the energy gained by relaxation on the excited state potential energy surface (PES). For this, the harmonic approximation needs to be applicable on the excited state PES of interest. The physical process that is described by the excited state JEDI analysis is fundamentally different from the ground state variant. While in the ground state JEDI analysis the distribution of stress energy in a mechanically deformed molecule is analyzed, i.e. energy is expended for deformation, the excited state JEDI analysis quantifies the energy gained by the relaxation of each internal mode upon relaxation on the excited state PES, i.e. energy becomes available. With the excited state JEDI analysis, the mechanical efficiency of molecular photoswitches can be calculated. The spatial extension of a photoswitch that undergoes, e.g., cis-trans-photoisomerization, changes significantly during this process and forces are exerted on the environment. However, other internal modes of the photoswitch that do not contribute to the change in spatial extension change as a side effect of the relaxation on the excited state PES and a certain amount of energy is wasted on them. This effect limits the mechanical efficiency of photoswitches. Using the excited state JEDI analysis, I investigated the mechanical efficiency of the stiff-stilbene photoswitch, which had been used in an experiment in literature to accelerate the electrocyclic ring opening of cyclobutene by photoisomerization. I found that the mechanical efficiency of stiff-stilbene is much too low to account for the observed enhancement of the reaction. A much more reasonable physical explanation is that excess energy from absorption of a photon is dissipated as heat, which accelerates the rupture of the thermally labile bond in cyclobutene. Furthermore, I used the JEDI analysis to investigate methods for the stabilization of strained hydrocarbons. Angle-strained cycloalkynes with a ring size smaller than eight carbon atoms are highly unstable under normal laboratory conditions, since the C≡C−C bond angles deviate substantially from linearity. Applying an external force in an appropriate direction partially linearizes these bond angles, thus leading to a stabilization of the cycloalkyne. Incorporating cycloheptyne into a macrocycle with stiff-stilbene, however, does not lead to a significant stabilization, since the mechanical efficiency of stiff-stilbene is low. Coupling cycloheptyne to another strained hydrocarbon, on the other hand, stabilizes the molecule tremendously. In particular, cycloheptyne was coupled to a strained cyclophane in a condensed macrocycle and I found that appropriate linking leads to a loss of strain in both hydrocarbons. In addition to the development of the JEDI force analysis tool, I used existing quantum mechanochemical methods to develop molecular force probes. This class of molecules can be incorporated into larger systems like polymers and proteins and can be used to monitor forces acting in different regions of the macromolecules in real-time via force-induced changes in the spectroscopic signals of the force probes. I found that the reduction of point group symmetry upon mechanical deformation of the molecular force probe is a profitable feature, since electronically excited states that are degenerate in the unperturbed state can split up upon application of an external force. This effect can lead to the generation of new peaks in the spectrum, thus allowing the precise identification of the force probe signal. Additionally, I incorporated molecular force probes into the backbone of proteins without disturbing their natural fold. The formation of hydrogen bonds between the force probe and neighboring strands in a β-sheet preserves the secondary and tertiary structure of the protein and allows the identification of the pulling direction. The application of forces along and perpendicular to the backbone yields pronounced and clearly distinguishable signals of the force probes in the infrared and Raman spectra. Advantageously, the intensities of these signals are proportional to the external force at selected points of the spectrum, which makes the molecules “force rulers”. The signals of the force probes can be intensified and shifted to a transparent window in the protein spectrum by isotopic substitution

Bending rigidity, supercoiling and knotting of ring polymers: models and simulations

The first part of the thesis was focussed on the interplay between knotting propensity and bending rigidity of equilibrated rings polymers. We found a surprising result: the equilibrium incidence of knots has a strongly non- monotonic dependence on bending, with a maximum at intermediate flexural rigidities. We next provided a quantitative framework, based on the balance of bending energy and configurational entropy, that allowed for rationalizing this counter-intuitive effect. We next extended the investigation to rings of much larger number of beads, via an heuristic model mapping between our semiflexible rings of beads and self-avoiding rings of cylinder. By the mapping, we not only confirmed the unimodal knotting profile for chains of 1,000 beads, but further found that chains of > 20,000 beads are expected to feature a bi-modal profile. We believe it would be most interesting to direct future efforts to confirm this transition from uni- to bi-modality using advanced sampling techniques for very long polymer rings. The second part of the thesis focused on the interplay of DNA knots and su- percoiling which are typically simultaneously present in vivo. We first studied this interplay by using oxDNA, an accurate mesoscopic DNA model and using it to study ings of thousands of base pairs tied in complex knots and with or without negative supercoiling (as appropriate for bacterial plasmids). By monitoring the dynamics of the DNA rings we found that the simultaneous presence of knots and supercoiling, and only their simultaneous presence, leads to a dramatic slowing down of the system reconfiguration dynamics. In particular, the essential tangles in the knotted region acquire a very long-lived character that, we speculate, could aid their recognition and simplification by topoisomerase. Finally, motivated by the recent experimental breakthrough that detected knots in eukaryotic DNA, we investigated the relationship between the compactness, writhe and knotting probability. The model was tuned to capture some of the salient properties of yeast minichromosomes, which were shown experimentally to become transiently highly knotted during transcription

Dynamics of DNA knots and links

The goal of this work is to describe the dynamics of DNA knots and links in an ionized fluid. To do so, we employ three models: 1. The Generalized Immersed Boundary (GIB) method, which is a deterministic method that accounts for the fluid, structure interaction of an immersed DNA molecule in an ionized fluid; 2. The Stochastic Generalized Boundary (SGIB) Method, which is an extension of the GIB method that also takes into account the random thermal fluctuations within the fluid; 3. The Sequence Dependent SGIB method, which is a new extension of the SGIB method that accounts for the elastic properties of a specified DNA sequence. Using the GIB and SGIB methods, we explore the energy landscape of a closed DNA segment in a trefoil knot configuration. We first analyze the symmetry of stable knotted equilibrium configurations, approximate saddle configurations, and examine elastic energy throughout the deterministic process. We then use the SGIB method to model DNA knot dynamics as a continuous time Markov chain. We classify and find boundaries within the energy landscape using Procrustes distance. Finally, we obtain a steady state distribution for the Markov process given a fixed linking number and compare this to the Gibb's distribution from energy estimates obtained from the GIB method. Lastly, using the SD-SGIB method, we also explore the effects of sequence dependence in the formation of kinetoplast DNA (kDNA), which has a chainmail-like linked DNA structure. We do so by finding the distribution of centroid distances of two kDNA minicircles

Multiscale structure and mechanics of collagen

While we are 70% water, in a very real sense collagen is the stuff we are made of. It is the most abundant protein in multicellular organisms, such as ourselves, making up roughly 25% of our total protein content. If you have ever wondered how the human body holds together all its different parts in shape, here is your answer: it is largely due to collagen. Collagen is the main ingredient of so called connective tissue which serves to hold the various parts of the body together. In fact, without collagen we would quite literally fall apart. Some genetic diseases, such as Osteogenesis Imperfecta (brittle bone disease) and Ehlers Danlos syndrome (characterized by abnormally stretchy skin and loose joints), are known to be the result of defective collagen. One can see why it is necessary to achieve a good understanding of how the strength or mechanical properties of collagen come about, and how it contributes to the state of well-being of an individual. This is essentially the aim of my research. Despite its relative abundance and about a century of research by many scientists, many features of collagen remain not fully understood. For example, it has been difficult to ascertain the precise structure of collagenous tissue. Nevertheless, a lot of progress has been made during the past couple of decades. Understanding the internal structure of collagen is important since we expect its strength to largely depend on its structure. The tools that physicists use to figure out the structure of biological tissue include advanced microscopes, such as the Atomic Force Microscope and the Electron Microscope, and X-ray diffraction analysis, in which one tries to determine the collagen structure at very small length scales (a billionth of a meter) by observing the fringe patterns made when an X-ray beam is scattered by the atoms that make up collagen (see Figure 1.2). These studies have revealed that there are 28 different types of collagens. The types are numbered with roman numerals I - XXVIII. In my work I have focused on the most common form of collagen, namely Type I collagen, which occurs mostly in scar tissue, bone and tendon. Tissue made from Type I collagen is made up of atoms that are arranged in such a way that a clear hierarchical structure emerges. This hierarchical organization is illustrated in Figure 1.1. The hierarchy can be described as follows: approximately 10-20 atoms are grouped together to form amino acids (there are 21 types of amino acids which, incidentally, nature also uses to encode our genetic information). About a 1000 amino acids of mainly two types, Glycine and Proline, are stringed together in a special repeating sequence to form polymers, or strands, called alpha helices (which, as the name implies, are shaped like helices, the shape of a spring) which are left-handed. Every three alpha helices associate with one another in a braided conformation that is a righthanded triple-helix called tropocollagen (see Figure 1.1 for an image)2. This is the basic building unit of collagen tissue. Certain cells of our bodies, called fibroblasts, are responsible for manufacturing tropocollagen molecules. Inside the body, after a fibroblast makes a tropocollagen molecule, it extrudes the molecule and aids in laying the tropocollagen molecule among other already existing tropocollagen molecules in order to forma long fine bundle known as a fibril. This process can also occur outside the body in the laboratory, unaided by fibroblasts but merely driven by thermal agitation at a particular range of temperatures. This is an example of a process known as self-assembly. The arrangement of tropocollagen molecules within a fibril is ‘staggered’, somewhat similar to the arrangement of bricks in a brick wall. In the body, many fibrils occur lying side by side and bundled together to form fibers which are then cross-linked to form part of the connective tissue. Special proteins, known as glycosaminoglycans (GAGs) are responsible for binding the fibrils together. For fibrils self-assembled outside the body and in the absence of GAGs, fibers do not form, but rather a network of fibrils with a well-defined diameter emerges. While the sequence of atoms that constitute collagen alpha-helices is precisely known, the precise arrangement of tropocollagen molecules within a fibril is difficult to ascertain by experimental means. This is because tropocollagen is a very light and flexible polymer, hence it is constantly changing its bent conformation in response to the erratic bombardment of fast moving atoms of the surrounding medium. This happens even within the closely packed environment of a fibril. Indeed, as the temperature of the surrounding medium increases, the atoms move even faster causing the tropocollagen molecule to wriggle even more. The consequence of this behaviour is that the molecule’s resistance to a stretching force increases with increasing temperature, just as for a rubber band when it is heated for example.3 Therefore the apparent randomness of the tropocollagen molecule’s wriggling form and motion affects the strength of collagenous tissue at its various levels within its hierarchical organization. In order to quantify this behaviour, special mathematical, or computational models, representing tropocollagen need to be proposed. Quantum Physics gives a precise mathematical description of the physical laws that govern how moving atoms interact with each other. One might think that the precise knowledge of all these atoms and how exactly they interact with each other should straightforwardly lead to the explanation and prediction of all possible phenomena in Nature. This may be true, in principle. However since all these interactions have been expressed in terms of mathematical theories and equations whose solutions can be difficult to compute, especially when such large numbers of atoms are simultaneously involved, physicists propose simplistic models called ‘coarse-grained models’ that are easier to manage computationally and that are assumed to encapsulate the essential properties of groups of these atoms. They go on to show by simulations of these models that the phenomena exhibited by these models do not necessarily have to depend on the internal details of the atoms they represent (nor on their interactions). In this thesis, we developed a coarse-grained model of the tropocollagen molecule that captures the essential features of tropocollagen. These features include its flexibility, its volume, and its tendency to stick to other tropocollagen molecules that come near it. We then generated an entire collagen fibril using many copies of this simplistic model of the tropocollagen molecule and attaching (or cross-linking) them to each other at specific points on the molecules. Then we simulated the entire cross-linked assembly of tropocollagen molecules on a computer at a particular temperature and then attempted to estimate the strength the fibril. In our coarse-grained model, each tropocollagen molecule was transformed into a chain of about 100 identical rigid sticks (called bonds) but balljointed at their ends to one another in a linear sequence (see Figure 7.1). The ease with which the joints could rotate depended on a single quantity known as the bending stiffness of tropocollagen, which has been measured by experiments in the laboratory. This model is called the ‘discrete worm-like chain’. Because in reality the tropocollagen molecule is constantly wriggling, it is difficult to follow precisely its full motion in time, even on a computer. So we rather chose a statistical treatment. This is where statistical physics and Monte Carlo methods become very useful. Simply put, Monte Carlo methods repeatedly generate sets of random numbers and use them to propose different configurations of the tropocollagen monomers every time. This is somewhat like throwing dice to obtain different numbers every time they are thrown. However, the Monte Carlo method that is employed should take care not to change the lengths of the bonds of the simplistic model, also it must never destroy any of the cross-links in the system. Prior to this work it has not been possible satisfy all these constraints during the simulation of an arbitrarily cross-linked assembly of coarse-grained polymers. In this thesis, however, we invented a Monte Carlo method that overcomes these issues. We named the method ‘TRACTRIX’, because it is based on the construction of a special curve which in mathematics is called a tractrix. The tractrix is the answer to the following question: "Given two points linked by a rigid joint, if one point moves along a given curve, how does the other point move?" (See Figure 4.3 for an example of how a tractrix looks like.) The details of this method are described in Chapter 4. In Chapter 5, we demonstrated that TRACTRIX works, that is, it is accurate and trustworthy, because it reproduced expected results for certain model polymer networks for which we already knew the exact answers. TRACTRIX was then used to simulate our model of the collagen fibril, and we were thus able to estimate the strength of the collagen fibril. Other interesting results were also established, such as the shape that the fibril finally settled into, and also the average conformation of its constituent tropocollagen molecules. Our model and simulations set the stage for further investigation into the mechanics and other properties of collagen fibrils. Also this exciting new method TRACTRIX promises to be useful in simulating many different types of biological networks, not only collagen