We report a novel simulation strategy that enables us to identify key parameters controlling the experimentally measurable characteristics of structural protein tags on dsDNA construct translocating through a double nanopore âŠ
We report a novel simulation strategy that enables us to identify key parameters controlling the experimentally measurable characteristics of structural protein tags on dsDNA construct translocating through a double nanopore setup. First, we validate the scheme in silico by reproducing and explaining the physical origin of the experimental dwell time distributions of the Streptavidin markers on a 48 kbp long dsDNA. These studies reveal the important differences in the characteristics of the protein tags compared to the dynamics of dsDNA segments, immediately providing clues on how to improve the measurement protocols to decipher the unknown genomic lengths accurately. Of particular importance is the in silico studies on the effect of electric field inside and beyond the pores which we find is critical to discriminate protein tags based on their effective charges and masses revealed through a generic power-law dependence of the average dwell time at each pore. The simulation protocols enable to monitor piecewise dynamics of the individual monomers at a sub-nanometer length scale and provide an explanation of the disparate velocity variation from one tag to the other using the nonequilibrium tension propagation theory, - a key element to decipher genomic lengths accurately. We further justify the model and the chosen simulation parameters by calculating the Peclet number which is in close agreement with the experiment. Analysis of our simulation results from the CG model has the capability to refine the accuracy of the experimentally obtained genomic lengths and carefully chosen simulation strategies can serve as a powerful tool to discriminate different types of neutral and charged tags of different origins on a dsDNA construct in terms of their physical characteristics and can provide insights to increase both the efficiency and accuracy of an experimental dual-nanopore setup.
Abstract Methods for reducing and directly controlling the speed of DNA through a nanopore are needed to enhance sensing performance for direct strand sequencing and detection/mapping of sequenceâspecific features. A âŠ
Abstract Methods for reducing and directly controlling the speed of DNA through a nanopore are needed to enhance sensing performance for direct strand sequencing and detection/mapping of sequenceâspecific features. A method is created for reducing and controlling the speed of DNA that uses two independently controllable nanopores operated with an active control logic. The pores are positioned sufficiently close to permit cocapture of a single DNA by both pores. Once cocapture occurs, control logic turns on constant competing voltages at the pores leading to a âtugâofâwarâ whereby opposing forces are applied to regions of the molecules threading through the pores. These forces exert both conformational and speed control over the cocaptured molecule, removing folds and reducing the translocation rate. When the voltages are tuned so that the electrophoretic force applied to both pores comes into balance, the life time of the tugâofâwar state is limited purely by diffusive sliding of the DNA between the pores. A tugâofâwar state is produced on 76.8% of molecules that are captured with a maximum twoâorder of magnitude increase in average pore translocation time relative to the average time for singleâpore translocation. Moreover, the translocation slowâdown is quantified as a function of voltage tuning and it is shown that the slowâdown is well described by a first passage analysis for a 1D subdiffusive process. The ionic current of each nanopore provides an independent sensor that synchronously measures a different region of the same molecule, enabling sequential detection of physical labels, such as monostreptavidin tags. With advances in devices and control logic, future dualâpore applications include genome mapping and enzymeâfree sequencing.
Methods for reducing and directly controlling the speed of DNA through a nanopore are needed to enhance sensing performance for direct strand sequencing and detection/mapping of sequence-specific features. We have âŠ
Methods for reducing and directly controlling the speed of DNA through a nanopore are needed to enhance sensing performance for direct strand sequencing and detection/mapping of sequence-specific features. We have created a method for reducing and controlling the speed of DNA that uses two independently controllable nanopores operated with an active control logic. The pores are positioned sufficiently close to permit co-capture of a single DNA by both pores. Once co-capture occurs, control logic turns on constant competing voltages at the pores leading to a `tug-of-war' whereby the molecule is pulled from both ends by opposing forces. These forces exert both conformational and speed control over the co-captured molecule, removing folds and reducing the translocation rate. When the voltages are tuned so that the electrophoretic force applied to both ends of the molecule comes into balance, the life-time of the tug-of-war state is limited purely by diffusive sliding of the DNA between the pores. We are able to produce a tug-of-war state on 76.8% of molecules that are captured with a maximum two-order of magnitude increase in average pore translocation time relative to the average time for single-pore translocation. Moreover, we quantify the translocation slow-down as a function of voltage tuning and show that the slow-down is well described by a first passage analysis for a one-dimensional sub-diffusive process. The ionic current of each nanopore provides an independent sensor that synchronously measures a different region of the same molecule, enabling sequential detection of physical labels, such as mono-streptavidin tags. With advances in devices and control logic, future dual-pore applications include genome mapping and enzyme-free sequencing.
The classical Stefan problem is a linear one-dimensional heat equation with a free boundary at one end, modelling a column of liquid (e.g. water) in contact with an infinite strip âŠ
The classical Stefan problem is a linear one-dimensional heat equation with a free boundary at one end, modelling a column of liquid (e.g. water) in contact with an infinite strip of solid (ice). Given the fixed boundary conditions, the column temperature and free boundary motion can be uniquely determined. In the inverse problem, one specifies the free boundary motion, say from one steady-state length to another, and seeks to determine the column temperature and fixed boundary conditions, or boundary control. This motion planning problem is a simplified version of a crystal growth problem. In this paper, we consider motion planning of the free boundary (Stefan) problem with a quadratic nonlinear reaction term. The treatment here is a first step towards treating higher order nonlinearities as observed in crystal growth furnaces. Convergence of a series solution is proven and a detailed parametric study on the series radius of convergence given. Moreover, we prove that the parametrization can indeed be used for motion planning purposes; computation of the open loop motion planning is straightforward and we give simulation results.
In this paper we consider a free boundary problem for a nonlinear parabolic partial differential equation. In particular, we are concerned with the inverse problem, which means we know the âŠ
In this paper we consider a free boundary problem for a nonlinear parabolic partial differential equation. In particular, we are concerned with the inverse problem, which means we know the behavior of the free boundary a priori and would like a solution, e.g. a convergent series, in order to determine what the trajectories of the system should be for steady-state to steady-state boundary control. In this paper we combine two issues: the free boundary (Stefan) problem with a quadratic nonlinearity. We prove convergence of a series solution and give a detailed parametric study on the series radius of convergence. Moreover, we prove that the parametrization can indeed can be used for motion planning purposes; computation of the open loop motion planning is straightforward. Simulation results are given and we prove some important properties about the solution. Namely, a weak maximum principle is derived for the dynamics, stating that the maximum is on the boundary. Also, we prove asymptotic positiveness of the solution, a physical requirement over the entire domain, as the transient time from one steady-state to another gets large.
General theory of ODE and Gevrey classes, Chen Hua and Luigi Rodino pseudo-differential operators - an introduction, Otto Liess coherent states and evolution equations, Nicolas Lerner non-linear microlocal analysis, Xu âŠ
General theory of ODE and Gevrey classes, Chen Hua and Luigi Rodino pseudo-differential operators - an introduction, Otto Liess coherent states and evolution equations, Nicolas Lerner non-linear microlocal analysis, Xu Chao-Jing propagation of sungularities and second microlocalization, Chen Shuxing analytic hyperbolic systems, Jean Vaillant.
We study the dynamics of the passage of a polymer through a membrane pore (translocation), focusing on the scaling properties with the number of monomers N. The natural coordinate for âŠ
We study the dynamics of the passage of a polymer through a membrane pore (translocation), focusing on the scaling properties with the number of monomers N. The natural coordinate for translocation is the number of monomers on one side of the hole at a given time. Commonly used models that assume Brownian dynamics for this variable predict a mean (unforced) passage time tau that scales as N2, even in the presence of an entropic barrier. In particular, however, the time it takes for a free polymer to diffuse a distance of the order of its radius by Rouse dynamics scales with an exponent larger than two, and this should provide a lower bound to the translocation time. To resolve this discrepancy, we perform numerical simulations with Rouse dynamics for both phantom (in space dimensions d=1 and 2), and self-avoiding (in d=2) chains. The results indicate that for large N, translocation times scale in the same manner as diffusion times, but with a larger prefactor that depends on the size of the hole. Such scaling implies anomalous dynamics for the translocation process. In particular, the fluctuations in the monomer number at the hole are predicted to be nondiffusive at short times, while the average pulling velocity of the polymer in the presence of a chemical-potential difference is predicted to depend on N.
We consider a one-phase Stefan problem for the heat equation with a superlinear reaction term. It is known from a previous work (Ghidouche, Souplet, & Tarzia [5]) that all global âŠ
We consider a one-phase Stefan problem for the heat equation with a superlinear reaction term. It is known from a previous work (Ghidouche, Souplet, & Tarzia [5]) that all global solutions are bounded and decay uniformly to 0. Moreover, it was shown in Ghidouche, Souplet, & Tarzia [5] that either: (i) the free boundary converges to a finite limit and the solution decays at an exponential rate, or (ii) the free boundary grows up to infinity and the decay rate is at most polynomial, and it was also proved that small data solutions behave like (i).
A new system equivalence relation, using the framework of differential geometry of jets and prolongations of infinite order, is studied. In this setting, two systems are said to be equivalent âŠ
A new system equivalence relation, using the framework of differential geometry of jets and prolongations of infinite order, is studied. In this setting, two systems are said to be equivalent if any variable of one system may be expressed as a function of the variables of the other system and of a finite number of their time derivatives. This is a Lie-Backlund isomorphism. The authors prove that, although the state dimension is not preserved, the number of input channels is kept fixed. They also prove that a Lie-Backlund isomorphism can be realized by an endogenous feedback. The differentially flat nonlinear systems introduced by the authors (1992) via differential algebraic techniques, are generalized and the new notion of orbitally flat systems is defined. They correspond to systems which are equivalent to a trivial one, with time preservation or not. The endogenous linearizing feedback is explicitly computed in the case of the VTOL aircraft to track given reference trajectories with stability.
Preface Prologue: The Exponential Function Chapter 1: Abstract Integration Set-theoretic notations and terminology The concept of measurability Simple functions Elementary properties of measures Arithmetic in [0, ] Integration of positive âŠ
Preface Prologue: The Exponential Function Chapter 1: Abstract Integration Set-theoretic notations and terminology The concept of measurability Simple functions Elementary properties of measures Arithmetic in [0, ] Integration of positive functions Integration of complex functions The role played by sets of measure zero Exercises Chapter 2: Positive Borel Measures Vector spaces Topological preliminaries The Riesz representation theorem Regularity properties of Borel measures Lebesgue measure Continuity properties of measurable functions Exercises Chapter 3: Lp-Spaces Convex functions and inequalities The Lp-spaces Approximation by continuous functions Exercises Chapter 4: Elementary Hilbert Space Theory Inner products and linear functionals Orthonormal sets Trigonometric series Exercises Chapter 5: Examples of Banach Space Techniques Banach spaces Consequences of Baire's theorem Fourier series of continuous functions Fourier coefficients of L1-functions The Hahn-Banach theorem An abstract approach to the Poisson integral Exercises Chapter 6: Complex Measures Total variation Absolute continuity Consequences of the Radon-Nikodym theorem Bounded linear functionals on Lp The Riesz representation theorem Exercises Chapter 7: Differentiation Derivatives of measures The fundamental theorem of Calculus Differentiable transformations Exercises Chapter 8: Integration on Product Spaces Measurability on cartesian products Product measures The Fubini theorem Completion of product measures Convolutions Distribution functions Exercises Chapter 9: Fourier Transforms Formal properties The inversion theorem The Plancherel theorem The Banach algebra L1 Exercises Chapter 10: Elementary Properties of Holomorphic Functions Complex differentiation Integration over paths The local Cauchy theorem The power series representation The open mapping theorem The global Cauchy theorem The calculus of residues Exercises Chapter 11: Harmonic Functions The Cauchy-Riemann equations The Poisson integral The mean value property Boundary behavior of Poisson integrals Representation theorems Exercises Chapter 12: The Maximum Modulus Principle Introduction The Schwarz lemma The Phragmen-Lindelof method An interpolation theorem A converse of the maximum modulus theorem Exercises Chapter 13: Approximation by Rational Functions Preparation Runge's theorem The Mittag-Leffler theorem Simply connected regions Exercises Chapter 14: Conformal Mapping Preservation of angles Linear fractional transformations Normal families The Riemann mapping theorem The class L Continuity at the boundary Conformal mapping of an annulus Exercises Chapter 15: Zeros of Holomorphic Functions Infinite Products The Weierstrass factorization theorem An interpolation problem Jensen's formula Blaschke products The Muntz-Szas theorem Exercises Chapter 16: Analytic Continuation Regular points and singular points Continuation along curves The monodromy theorem Construction of a modular function The Picard theorem Exercises Chapter 17: Hp-Spaces Subharmonic functions The spaces Hp and N The theorem of F. and M. Riesz Factorization theorems The shift operator Conjugate functions Exercises Chapter 18: Elementary Theory of Banach Algebras Introduction The invertible elements Ideals and homomorphisms Applications Exercises Chapter 19: Holomorphic Fourier Transforms Introduction Two theorems of Paley and Wiener Quasi-analytic classes The Denjoy-Carleman theorem Exercises Chapter 20: Uniform Approximation by Polynomials Introduction Some lemmas Mergelyan's theorem Exercises Appendix: Hausdorff's Maximality Theorem Notes and Comments Bibliography List of Special Symbols Index
In this short paper, a correction is made to the recently proposed solution of Li and Talaga to a 1D biased diffusion model for linear DNA translocation, and a new âŠ
In this short paper, a correction is made to the recently proposed solution of Li and Talaga to a 1D biased diffusion model for linear DNA translocation, and a new analysis will be given to their data. It was pointed out by us recently that this 1D linear translocation model is equivalent to the one that was considered by Schrödinger for the EhrenhaftâMillikan measurements on electron charge. Here, we apply Schrödingerâs first-passage-time distribution formula to the data set in Li and Talaga. It is found that Schrödingerâs formula can be used to describe the time distribution of DNA translocation in solid-state nanopores. These fittings yield two useful parameters: the drift velocity of DNA translocation and the diffusion constant of DNA inside the nanopore. The results suggest two regimes of DNA translocation: (I) at low voltages, there are clear deviations from Smoluchowskiâs linear law of electrophoresis, which we attribute to the entropic barrier effects; (II) at high voltages, the translocation velocity is a linear function of the applied electric field. In regime II, the apparent diffusion constant exhibits a quadratic dependence on the applied electric field, suggesting a mechanism of Taylor-dispersion effect likely due the electro-osmotic flow field in the nanopore channel. This analysis yields a dispersion-free diffusion constant value of 11.2 nm2 ”s-1 for the segment of DNA inside the nanopore, which is in quantitative agreement with the StokesâEinstein theory. The implication of Schrödingerâs formula for DNA sequencing is discussed.
In this paper we consider a free boundary problem for a nonlinear parabolic partial differential equation. In particular, we are concerned with the inverse problem, which means we know the âŠ
In this paper we consider a free boundary problem for a nonlinear parabolic partial differential equation. In particular, we are concerned with the inverse problem, which means we know the behavior of the free boundary a priori and would like a solution, e.g. a convergent series, in order to determine what the trajectories of the system should be for steady-state to steady-state boundary control. In this paper we combine two issues: the free boundary (Stefan) problem with a quadratic nonlinearity. We prove convergence of a series solution and give a detailed parametric study on the series radius of convergence. Moreover, we prove that the parametrization can indeed can be used for motion planning purposes; computation of the open loop motion planning is straightforward. Simulation results are given and we prove some important properties about the solution. Namely, a weak maximum principle is derived for the dynamics, stating that the maximum is on the boundary. Also, we prove asymptotic positiveness of the solution, a physical requirement over the entire domain, as the transient time from one steady-state to another gets large.
We consider a model system in which anomalous diffusion is generated by superposition of underlying linear modes with a broad range of relaxation times. In the language of Gaussian polymers, âŠ
We consider a model system in which anomalous diffusion is generated by superposition of underlying linear modes with a broad range of relaxation times. In the language of Gaussian polymers, our model corresponds to Rouse (Fourier) modes whose friction coefficients scale as wavenumber to the power $2-z$. A single (tagged) monomer then executes subdiffusion over a broad range of time scales, and its mean square displacement increases as $t^\alpha$ with $\alpha=1/z$. To demonstrate non-trivial aspects of the model, we numerically study the absorption of the tagged particle in one dimension near an absorbing boundary or in the interval between two such boundaries. We obtain absorption probability densities as a function of time, as well as the position-dependent distribution for unabsorbed particles, at several values of $\alpha$. Each of these properties has features characterized by exponents that depend on $\alpha$. Characteristic distributions found for different values of $\alpha$ have similar qualitative features, but are not simply related quantitatively. Comparison of the motion of translocation coordinate of a polymer moving through a pore in a membrane with the diffusing tagged monomer with identical $\alpha$ also reveals quantitative differences.
When a flexible polymer is sucked into a localized small hole, the chain can initially respond only locally and the sequential nonequilibrium processes follow in line with the propagation of âŠ
When a flexible polymer is sucked into a localized small hole, the chain can initially respond only locally and the sequential nonequilibrium processes follow in line with the propagation of the tensile force along the chain backbone. We analyze this dynamical process by taking the nonuniform stretching of the polymer into account both with and without hydrodynamics interactions. Earlier conjectures on the absorption time are criticized and formulas are proposed together with time evolutions of relevant dynamical variables.
Motivated by a novel method for granular segregation, we analyse the one-dimensional drift-diffusion between two absorbing boundaries. The time evolution of the probability distribution and the rate of absorption are âŠ
Motivated by a novel method for granular segregation, we analyse the one-dimensional drift-diffusion between two absorbing boundaries. The time evolution of the probability distribution and the rate of absorption are given by explicit formulae; the splitting probability and the mean first-passage time are also calculated. Applying the results we find optimal parameters for segregating binary granular mixtures.
We review recent progress on the theory of dynamics of polymer translocation through a nanopore based on the iso-flux tension propagation (IFTP) theory. We investigate both pore-driven translocation of flexible âŠ
We review recent progress on the theory of dynamics of polymer translocation through a nanopore based on the iso-flux tension propagation (IFTP) theory. We investigate both pore-driven translocation of flexible and a semi-flexible polymers, and the end-pulled case of flexible chains by means of the IFTP theory and extensive molecular dynamics (MD) simulations. The validity of the IFTP theory can be quantified by the waiting time distributions of the monomers which reveal the details of the dynamics of the translocation process. The IFTP theory allows a parameter-free description of the translocation process and can be used to derive exact analytic scaling forms in the appropriate limits, including the influence due to the pore friction that appears as a finite-size correction to asymptotic scaling. We show that in the case of pore-driven semi-flexible and end-pulled polymer chains the IFTP theory must be augmented with an explicit trans side friction term for a quantitative description of the translocation process.
This book is of interest to mathematicians and computer scientists working in finite mathematics and combinatorics. It presents a breakthrough method for analyzing complex summations. Beautifully written, the book contains âŠ
This book is of interest to mathematicians and computer scientists working in finite mathematics and combinatorics. It presents a breakthrough method for analyzing complex summations. Beautifully written, the book contains practical applications as well as conceptual developments that will have applications in other areas of mathematics.From the ta
We suggest a governing equation that describes the process of polymer-chain translocation through a narrow pore and reconciles the seemingly contradictory features of such dynamics: (i) a Gaussian probability distribution âŠ
We suggest a governing equation that describes the process of polymer-chain translocation through a narrow pore and reconciles the seemingly contradictory features of such dynamics: (i) a Gaussian probability distribution of the translocated number of polymer segments at time t after the process has begun, and (ii) a subdiffusive increase of the distribution variance Î(t) with elapsed time Î(t)ât(α). The latter quantity measures the mean-squared number s of polymer segments that have passed through the pore Î(t)=([s(t)-s(t=0)](2)), and is known to grow with an anomalous diffusion exponent α<1. Our main assumption [i.e., a Gaussian distribution of the translocation velocity v(t)] and some important theoretical results, derived recently, are shown to be supported by extensive Brownian dynamics simulation, which we performed in 3D. We also numerically confirm the predictions made recently that the exponent α changes from 0.91 to 0.55 to 0.91 for short-, intermediate-, and long-time regimes, respectively.