Quantifying Intermittent Transport in Cell Cytoplasm
|
|
- Charles McCormick
- 6 years ago
- Views:
Transcription
1 Quantifying Intermittent Transport in Cell Cytoplasm Ecole Normale Supérieure, Mathematics and Biology Department. Paris, France. May 19 th 2009
2 Cellular Transport Introduction Cellular Transport Intermittent Search Mechanism Early steps of viral infection Scheme Motivations Extra and Intracellular communication Intermittent transport: diffusion and active motion alternation Active motion along microtubules (MTs) via molecular motors
3 Intermittent Search Mechanism Cellular Transport Intermittent Search Mechanism Early steps of viral infection Scheme Motivations Alternation between diffusion and directed motion to a target mrna granules to synaptic targets along a dendrite. DNA-viruses to nuclear pores
4 Early steps of viral infection Cellular Transport Intermittent Search Mechanism Early steps of viral infection Scheme Motivations 1-2: extracellular diffusion and membrane exploring 3: Entry 3-4: Intermittent transport: diffusion and directed motion along MTs 4: Nuclear delivery of DNA Figure: G. Seisengerger et al., Science 294, 1929 (2001).
5 Scheme Introduction Cellular Transport Intermittent Search Mechanism Early steps of viral infection Scheme Motivations
6 Motivations Introduction Cellular Transport Intermittent Search Mechanism Early steps of viral infection Scheme Motivations Deriving drift accounting for intermittent transport Langevin description of trajectories Application to viral infection analysis: possible degradation in cytoplasm Mean Time τ e and Probability P e a virus enters a nuclear pore?
7 Langevin Description Langevin Description of Trajectories Fokker-Planck Equation Probality P e and mean time τ e to a nuclear pore Asymptotic Results Left-Hand side: Intermittent Dynamics ẋ = 2Dẇ Free Particle, ẋ = V Bound Particle. Right-Hand side: Langevin Dynamics ẋ = b(x) + 2Dẇ +killing field k(x)
8 Fokker-Planck Equation Langevin Description of Trajectories Fokker-Planck Equation Probality P e and mean time τ e to a nuclear pore Asymptotic Results Survival probability:p(x, y, t) = Pr{X (t) x + dx X (0) = y } Forward Fokker-Planck Equation t p = D p (p b (x)) k (x) p boundary conditions: p = 0 on Ω a (nuclear pores) and p n = 0 on Ω Ω a.
9 Langevin Description of Trajectories Fokker-Planck Equation Probality P e and mean time τ e to a nuclear pore Asymptotic Results Probality P e and mean time τ e to a nuclear pore P e and τ e P e = 1 τ e = 0 0 Ω k(x) p(x, t)dxdt k(x)t p(x, t)dxdt Ω p(x, t)dxdt 0 where p(x, t) = Ω p(x, y, t)p i(y)dy P e Ω
10 Asymptotic Results Langevin Description of Trajectories Fokker-Planck Equation Probality P e and mean time τ e to a nuclear pore Asymptotic Results Nuclear pores ( Ω a )= small holes Ωa Ω = ɛ 1 Asymptotic Results in ɛ P e = ln( 1 ɛ) Dπ Φ(x) 1 Ω R Ω e D ds x, Φ(x) RΩ e D k(x)dx+ 1 Φ(x) Ω R Ω e D ds x τ e = ln ( 1 ɛ ) Dπ ln ( 1 ɛ ) Dπ RΩ Φ(x) e D dx, Φ(x) RΩ e D k(x)dx+ 1 Φ(x) Ω R Ω e D ds x for b = Φ
11 Asymptotic Results Langevin Description of Trajectories Fokker-Planck Equation Probality P e and mean time τ e to a nuclear pore Asymptotic Results Nuclear pores ( Ω a )= small holes Ωa Ω = ɛ 1 Asymptotic Results in ɛ P e = ln( 1 ɛ) Dπ Φ(x) 1 Ω R Ω e D ds x, Φ(x) RΩ e D k(x)dx+ 1 Φ(x) Ω R Ω e D ds x τ e = ln ( 1 ɛ ) Dπ ln ( 1 ɛ ) Dπ RΩ Φ(x) e D dx, Φ(x) RΩ e D k(x)dx+ 1 Φ(x) Ω R Ω e D ds x for b = Φ PROBLEM: b?
12 Principle Introduction Principle Cell Representation Two-dimensional radial case Cylindrical neurite case MFPTs from x 0 to x f are equal. In the small diffusion limit: x f x 0 b(x 0 ) = τ(x 0 ) + t m
13 Cell representation Introduction Two-dimensional radial cell with N uniformly distributed microtubules: Principle Cell Representation Two-dimensional radial case Cylindrical neurite case Neurite cross section with N thin cylindrical MTs
14 Two-dimensional representation Principle Cell Representation Two-dimensional radial case Cylindrical neurite case In the small diffusion limit r 0 r f b(r 0 ) = r 0 ( r(r 0 ) d m ) = τ(r 0 ) + t m b(r 0 )
15 MFPT to a microtubule Principle Cell Representation Two-dimensional radial case Cylindrical neurite case Reflecting boundary!" r virus Dynkin s system!" a binding site #!" a brownian motion R D u(r, θ) = 1 in Ω u(r, 0) = u(r, Θ) = 0, u (R, θ) r = 0. Absorbing boundary For Θ << 1 τ(r 0 ) = 1 Θ Θ 0 u(r 0, θ)dθ r 2 0 Θ 2 12D
16 Mean binding radius (1) Principle Cell Representation Two-dimensional radial case Cylindrical neurite case Heat equation D p(r, θ, t) = p (r, θ, t) in Ω t p(r, 0, t) = p(r, Θ), t = 0, p (R, θ, t) r = 0.
17 Mean binding radius (1) Principle Cell Representation Two-dimensional radial case Cylindrical neurite case Heat equation Indeed, r(r 0 ) = 1 Θ D p(r, θ, t) = p (r, θ, t) in Ω t p(r, 0, t) = p(r, Θ), t = 0, p (R, θ, t) r = 0. Θ R 0 0 rɛ(r r 0, θ 0 )dθ 0 with ɛ(r r 0, θ 0 ) = 0 j(r, t r 0, θ 0 )dt = D p 0 n (r, t r 0, θ 0 )dt.
18 Mean binding radius (2) Principle Cell Representation Two-dimensional radial case Cylindrical neurite case Exit radius distribution Dotted line: Theoretical exit radius distribution Solid line: Numerical distribution (Brownian trajectories) initial radius r0= Radius For Θ << 1 r(r 0 ) r 0 (1 + Θ2 12 )
19 Results Introduction Principle Cell Representation Two-dimensional radial case Cylindrical neurite case Effective drift amplitude b(r 0 ) = r 0 ( r(r 0 ) d m ) = d Θ m r τ(r 0 ) + t m t m + r0 2 Θ2 12D ( ) Φ(r) = dm 12Dtm t mθ arctan Θr 12Dtm D 2 ln ( 12Dt m + r 2 Θ 2)
20 Results Introduction Principle Cell Representation Two-dimensional radial case Cylindrical neurite case Effective drift amplitude b(r 0 ) = r 0 ( r(r 0 ) d m ) = d Θ m r τ(r 0 ) + t m t m + r0 2 Θ2 12D ( ) Φ(r) = dm 12Dtm t mθ arctan Θr 12Dtm D 2 ln ( 12Dt m + r 2 Θ 2) Steady State Distribution Solid line: Numerical Distribution (intermittent Brownian trajectories) Dotted line: theoretical distribution obtained with Langevin description Radius Quantifying (µm) Intermittent Transport in Cell Cytoplasm
21 Cylindrical neurite case Principle Cell Representation Two-dimensional radial case Cylindrical neurite case Cross section of a neurite In the small diffusion limit b = d m t m + τ with τ 1 λ 1 = Ω ln( 1 ɛ) 2πN the MFPT to a microtubule.
22 Results(1) with the two-dimensional potential ( ) Φ(r) = dm 12Dtm t mθ arctan Θr 12Dtm D 2 ln ( 12Dt m + r 2 Θ 2)
23 Results(1) with the two-dimensional potential ( ) Φ(r) = dm 12Dtm t mθ arctan Θr 12Dtm D 2 ln ( 12Dt m + r 2 Θ 2) Probability and mean time to a nuclear pore P e τ e ( d m d m + K K k (d m + K) where K = 2k 0 δt m ln ( 1 ɛ 1 Kδ (d ) mδ + Dt m ) 12Dt m d m (d m + K) Θ2 ( 1 + δ (d ) mδ + Dt m ) 12Dt m (d m + K) Θ2. ) ( ) and α = 1 + R+δ 1 d m 24.
24 Results (2) with biological data: Probability and mean time to a nuclear pore P e 95% τ e 3min.
25 Results (2) with biological data: Probability and mean time to a nuclear pore P e 95% τ e 3min. coherent with the reported total entry time of 15min. (G. Seisengerger et al., Science 294, 1929 (2001)).
26 Results (2) with biological data: Probability and mean time to a nuclear pore P e 95% τ e 3min. coherent with the reported total entry time of 15min. (G. Seisengerger et al., Science 294, 1929 (2001)). without drift: τ e 15min.
27 General framework to analyze intermittent search processes Application to viral entry modelling
28 Introduction Asymptotics for structured targets Other steps of viral infection (endosome escape... ) Asymptotics for structured targets (many nuclear pores on a spherical nuclear pore... )
29 Asymptotics for structured targets Asymptotics for structured targets (pure diffusion b = 0) n disks (nuclear pores) of radius η located on a microdomain (capacitance C S : for a spherical nucleus of radius δ, C S = 4πδ) Old Asymptotics τ e = 1 + ( Ω 4Dnη ( R ) Ω k(x)dx 4Dnη Problem: lim n,nɛ 2 1 τ e = 0 ) New Asymptotics τ e = 1 + ( Ω D C ( R ) Ω k(x)dx D C where 1 C 1 C S + 1 4nη ) New asymptotics with a drift??
30 Introduction Asymptotics for structured targets The lab
31 Negative drift Introduction Asymptotics for structured targets Noise due to reflecting external membrane Steady state distribution Dashed line: Theoretical Langevin distribution Solid line: Intermittent Brownian simulations Radius (µm)
32 Limit radius Introduction Asymptotics for structured targets In cell of radius 50µm, positive drift for d m 1µm
33 Escape through a small hole (1) Asymptotics for structured targets How long it takes for a brownian particle confined to a domain Ω to escape through a small opening Ω a (ɛ = Ωa Ω << 1)? Mean escape time τ = Ω πd ln τ = Ω 4ɛD ( ) 1 ɛ (2-dimensional case), (3-dimensional case),
34 Escape through a small hole (2) Dynkin s system Asymptotics for structured targets u(x) = 1 D in Ω Neumann Function N (x, ξ) u(x) = 0 on Ω a u n (x) = 0 on Ω r = Ω Ω a. N (x, ξ) = δ(x ξ) for x, ξ Ω N 1 (x, ξ) = for x Ω, ξ Ω. n Ω
35 Escape through a small hole (3) Asymptotics for structured targets Ω and N (x, ξ) u(x) N (x, ξ)u(x)dx = Ω N (x, ξ) u(x) N (x, ξ)u(x)dx = u(ξ) 1 D thus u(ξ) 1 D Ω + N (x, ξ) u Ω a n (x)dx 1 u(x)dx Ω Ω Ω N (x, ξ)dx N (x, ξ)dx = N (x, ξ) u Ω a n (x)dx + 1 Ω Ω u(x)dx
36 Escape through a small hole (4) Asymptotics for structured targets For ξ Ω a, C 0 the constant leading order in ɛ of u(x) and g(s) = g 0 u ɛ the local expansion of on the boundary: 2 s 2 1 N (x, ξ)dx = N (s)g(s)ds + C 0 D Ω Ω a N (s) = 1 4π s + regular function, 1 D Ω N (x, ξ)dx is bounded and g 0 = (compatibility condition). Thus: Ω 2πɛD n u(x) C 0 = Ω 4ɛD
David Holcman Zeev Schuss. Stochastic Narrow Escape in Molecular and Cellular Biology. Analysis and Applications
David Holcman Zeev Schuss Stochastic Narrow Escape in Molecular and Cellular Biology Analysis and Applications Stochastic Narrow Escape in Molecular and Cellular Biology David Holcman Zeev Schuss Stochastic
More informationMathematical Methods for Neurosciences. ENS - Master MVA Paris 6 - Master Maths-Bio ( )
Mathematical Methods for Neurosciences. ENS - Master MVA Paris 6 - Master Maths-Bio (2014-2015) Etienne Tanré - Olivier Faugeras INRIA - Team Tosca November 26th, 2014 E. Tanré (INRIA - Team Tosca) Mathematical
More informationThe purpose of this lecture is to present a few applications of conformal mappings in problems which arise in physics and engineering.
Lecture 16 Applications of Conformal Mapping MATH-GA 451.001 Complex Variables The purpose of this lecture is to present a few applications of conformal mappings in problems which arise in physics and
More informationDiffusion in the cell
Diffusion in the cell Single particle (random walk) Microscopic view Macroscopic view Measuring diffusion Diffusion occurs via Brownian motion (passive) Ex.: D = 100 μm 2 /s for typical protein in water
More informationStatistical Mechanics of Active Matter
Statistical Mechanics of Active Matter Umberto Marini Bettolo Marconi University of Camerino, Italy Naples, 24 May,2017 Umberto Marini Bettolo Marconi (2017) Statistical Mechanics of Active Matter 2017
More informationarxiv: v1 [cond-mat.stat-mech] 28 Apr 2018
Extreme Narrow escape: shortest paths for the first particles to escape through a small window arxiv:84.88v [cond-mat.stat-mech] 28 Apr 28 K. Basnayake, A. Hubl, Z. Schuss 2, D. Holcman Ecole Normale Supérieure,
More informationMATH 3150: PDE FOR ENGINEERS FINAL EXAM (VERSION D) 1. Consider the heat equation in a wire whose diffusivity varies over time: u k(t) 2 x 2
MATH 35: PDE FOR ENGINEERS FINAL EXAM (VERSION D). Consider the heat equation in a wire whose diffusivity varies over time: u t = u k(t) x where k(t) is some positive function of time. Assume the wire
More informationPath integrals for classical Markov processes
Path integrals for classical Markov processes Hugo Touchette National Institute for Theoretical Physics (NITheP) Stellenbosch, South Africa Chris Engelbrecht Summer School on Non-Linear Phenomena in Field
More informationMath 46, Applied Math (Spring 2009): Final
Math 46, Applied Math (Spring 2009): Final 3 hours, 80 points total, 9 questions worth varying numbers of points 1. [8 points] Find an approximate solution to the following initial-value problem which
More information5 Applying the Fokker-Planck equation
5 Applying the Fokker-Planck equation We begin with one-dimensional examples, keeping g = constant. Recall: the FPE for the Langevin equation with η(t 1 )η(t ) = κδ(t 1 t ) is = f(x) + g(x)η(t) t = x [f(x)p
More informationBrownian Motion: Fokker-Planck Equation
Chapter 7 Brownian Motion: Fokker-Planck Equation The Fokker-Planck equation is the equation governing the time evolution of the probability density of the Brownian particla. It is a second order differential
More informationSmoluchowski Diffusion Equation
Chapter 4 Smoluchowski Diffusion Equation Contents 4. Derivation of the Smoluchoswki Diffusion Equation for Potential Fields 64 4.2 One-DimensionalDiffusoninaLinearPotential... 67 4.2. Diffusion in an
More informationUsing Green s functions with inhomogeneous BCs
Using Green s functions with inhomogeneous BCs Using Green s functions with inhomogeneous BCs Surprise: Although Green s functions satisfy homogeneous boundary conditions, they can be used for problems
More informationStochastic Narrow Escape in Molecular and Cellular Biology
Stochastic Narrow Escape in Molecular and Cellular Biology David Holcman Zeev Schuss Stochastic Narrow Escape in Molecular and Cellular Biology Analysis and Applications 123 David Holcman Group of Applied
More informationA path integral approach to the Langevin equation
A path integral approach to the Langevin equation - Ashok Das Reference: A path integral approach to the Langevin equation, A. Das, S. Panda and J. R. L. Santos, arxiv:1411.0256 (to be published in Int.
More informationThe Smoluchowski-Kramers Approximation: What model describes a Brownian particle?
The Smoluchowski-Kramers Approximation: What model describes a Brownian particle? Scott Hottovy shottovy@math.arizona.edu University of Arizona Applied Mathematics October 7, 2011 Brown observes a particle
More informationMajor Concepts Kramers Turnover
Major Concepts Kramers Turnover Low/Weak -Friction Limit (Energy Diffusion) Intermediate Regime bounded by TST High/Strong -Friction Limit Smoluchovski (Spatial Diffusion) Fokker-Planck Equation Probability
More informationc 2016 SIAM. Published by SIAM under the terms
SIAM J. APPL. MATH. Vol. 76, No. 1, pp. 368 390 c 2016 SIAM. Published by SIAM under the terms of the Creative Commons 4.0 license REACTIVE BOUNDARY CONDITIONS AS LIMITS OF INTERACTION POTENTIALS FOR BROWNIAN
More informationBioNewMetrics and stochastic analysis of superresolution. David Holcman Ecole Normale Superieure
BioNewMetrics and stochastic analysis of superresolution trajectories David Holcman Ecole Normale Superieure What is Bionewmetrics? www.bionewmetrics.org Founded in 2009: Mathematical methods applied to
More informationIntegration by Substitution
November 22, 2013 Introduction 7x 2 cos(3x 3 )dx =? 2xe x2 +5 dx =? Chain rule The chain rule: d dx (f (g(x))) = f (g(x)) g (x). Use the chain rule to find f (x) and then write the corresponding anti-differentiation
More informationExit times of diffusions with incompressible drifts
Exit times of diffusions with incompressible drifts Andrej Zlatoš University of Chicago Joint work with: Gautam Iyer (Carnegie Mellon University) Alexei Novikov (Pennylvania State University) Lenya Ryzhik
More informationAsymptotic Analysis of First Passage Time Problems Inspired by Ecology
Bulletin of Mathematical Biology manuscript No. (will be inserted by the editor) Venu Kurella Justin C. Tzou Daniel Coombs Michael J. Ward Asymptotic Analysis of First Passage Time Problems Inspired by
More informationMATH 131P: PRACTICE FINAL SOLUTIONS DECEMBER 12, 2012
MATH 3P: PRACTICE FINAL SOLUTIONS DECEMBER, This is a closed ook, closed notes, no calculators/computers exam. There are 6 prolems. Write your solutions to Prolems -3 in lue ook #, and your solutions to
More informationIntroduction to nonequilibrium physics
Introduction to nonequilibrium physics Jae Dong Noh December 18, 2016 Preface This is a note for the lecture given in the 2016 KIAS-SNU Physics Winter Camp which is held at KIAS in December 17 23, 2016.
More informationPhysics 325: General Relativity Spring Final Review Problem Set
Physics 325: General Relativity Spring 2012 Final Review Problem Set Date: Friday 4 May 2012 Instructions: This is the third of three review problem sets in Physics 325. It will count for twice as much
More informationPhysique de la Matière Condensée, Ecole Polytechnique, Palaiseau, France
Denis Grebenkov a and Gleb Oshanin b a Physique de la Matière Condensée, Ecole Polytechnique, Palaiseau, France b Physique Théorique de la Matière Condensée (UMR CNRS) Sorbonne Universités Université Pierre
More informationLecture 1: Random walk
Lecture : Random walk Paul C Bressloff (Spring 209). D random walk q p r- r r+ Figure 2: A random walk on a D lattice. Consider a particle that hops at discrete times between neighboring sites on a one
More information1 Introduction. 2 Diffusion equation and central limit theorem. The content of these notes is also covered by chapter 3 section B of [1].
1 Introduction The content of these notes is also covered by chapter 3 section B of [1]. Diffusion equation and central limit theorem Consider a sequence {ξ i } i=1 i.i.d. ξ i = d ξ with ξ : Ω { Dx, 0,
More information16. Working with the Langevin and Fokker-Planck equations
16. Working with the Langevin and Fokker-Planck equations In the preceding Lecture, we have shown that given a Langevin equation (LE), it is possible to write down an equivalent Fokker-Planck equation
More informationStochastic Viral Dynamics with Beddington-DeAngelis Functional Response
Stochastic Viral Dynamics with Beddington-DeAngelis Functional Response Junyi Tu, Yuncheng You University of South Florida, USA you@mail.usf.edu IMA Workshop in Memory of George R. Sell June 016 Outline
More informationWeak Ergodicity Breaking. Manchester 2016
Weak Ergodicity Breaking Eli Barkai Bar-Ilan University Burov, Froemberg, Garini, Metzler PCCP 16 (44), 24128 (2014) Akimoto, Saito Manchester 2016 Outline Experiments: anomalous diffusion of single molecules
More informationLegendre s Equation. PHYS Southern Illinois University. October 13, 2016
PHYS 500 - Southern Illinois University October 13, 2016 PHYS 500 - Southern Illinois University Legendre s Equation October 13, 2016 1 / 10 The Laplacian in Spherical Coordinates The Laplacian is given
More information2012 NCTS Workshop on Dynamical Systems
Barbara Gentz gentz@math.uni-bielefeld.de http://www.math.uni-bielefeld.de/ gentz 2012 NCTS Workshop on Dynamical Systems National Center for Theoretical Sciences, National Tsing-Hua University Hsinchu,
More informationWeek 2 Notes, Math 865, Tanveer
Week 2 Notes, Math 865, Tanveer 1. Incompressible constant density equations in different forms Recall we derived the Navier-Stokes equation for incompressible constant density, i.e. homogeneous flows:
More informationFractional Laplacian
Fractional Laplacian Grzegorz Karch 5ème Ecole de printemps EDP Non-linéaire Mathématiques et Interactions: modèles non locaux et applications Ecole Supérieure de Technologie d Essaouira, du 27 au 30 Avril
More informationEffective dynamics for the (overdamped) Langevin equation
Effective dynamics for the (overdamped) Langevin equation Frédéric Legoll ENPC and INRIA joint work with T. Lelièvre (ENPC and INRIA) Enumath conference, MS Numerical methods for molecular dynamics EnuMath
More informationEvolution equations with spectral methods: the case of the wave equation
Evolution equations with spectral methods: the case of the wave equation Jerome.Novak@obspm.fr Laboratoire de l Univers et de ses Théories (LUTH) CNRS / Observatoire de Paris, France in collaboration with
More information1. Consider the 1-DOF system described by the equation of motion, 4ẍ+20ẋ+25x = f.
Introduction to Robotics (CS3A) Homework #6 Solution (Winter 7/8). Consider the -DOF system described by the equation of motion, ẍ+ẋ+5x = f. (a) Find the natural frequency ω n and the natural damping ratio
More informationMath 46, Applied Math (Spring 2008): Final
Math 46, Applied Math (Spring 2008): Final 3 hours, 80 points total, 9 questions, roughly in syllabus order (apart from short answers) 1. [16 points. Note part c, worth 7 points, is independent of the
More informationMATH 3150: PDE FOR ENGINEERS FINAL EXAM (VERSION A)
MAH 35: PDE FOR ENGINEERS FINAL EXAM VERSION A). Draw the graph of 2. y = tan x labelling all asymptotes and zeros. Include at least 3 periods in your graph. What is the period of tan x? See figure. Asymptotes
More informationProblems in diffusion and absorption: How fast can you hit a target with a random walk?
Problems in diffusion and absorption: How fast can you hit a target with a random walk? Andrew J. Bernoff Harvey Mudd College In collaboration with Alan Lindsay (Notre Dame) Thanks to Alan Lindsay, Michael
More informationSimulation Video. Sarafa Iyaniwura, Kelly Huang, Justin Tzou, and Colin Macdonald MFPT for moving (UBC) trap June 12, / 30
Simulation Video Sarafa Iyaniwura, Kelly Huang, Justin Tzou, and Colin Macdonald MFPT for moving (UBC) trap June 12, 2017 1 / 30 Mean First Passage Time for a small periodic moving trap inside a reflecting
More informationFokker-Planck Equation with Detailed Balance
Appendix E Fokker-Planck Equation with Detailed Balance A stochastic process is simply a function of two variables, one is the time, the other is a stochastic variable X, defined by specifying: a: the
More informationMath 31CH - Spring Final Exam
Math 3H - Spring 24 - Final Exam Problem. The parabolic cylinder y = x 2 (aligned along the z-axis) is cut by the planes y =, z = and z = y. Find the volume of the solid thus obtained. Solution:We calculate
More informationChemistry 532 Practice Final Exam Fall 2012 Solutions
Chemistry 53 Practice Final Exam Fall Solutions x e ax dx π a 3/ ; π sin 3 xdx 4 3 π cos nx dx π; sin θ cos θ + K x n e ax dx n! a n+ ; r r r r ˆL h r ˆL z h i φ ˆL x i hsin φ + cot θ cos φ θ φ ) ˆLy i
More informationMath 162: Calculus IIA
Math 62: Calculus IIA Final Exam ANSWERS December 9, 26 Part A. (5 points) Evaluate the integral x 4 x 2 dx Substitute x 2 cos θ: x 8 cos dx θ ( 2 sin θ) dθ 4 x 2 2 sin θ 8 cos θ dθ 8 cos 2 θ cos θ dθ
More informationFrom a Mesoscopic to a Macroscopic Description of Fluid-Particle Interaction
From a Mesoscopic to a Macroscopic Description of Fluid-Particle Interaction Carnegie Mellon University Center for Nonlinear Analysis Working Group, October 2016 Outline 1 Physical Framework 2 3 Free Energy
More informationSummer 2017 MATH Solution to Exercise 5
Summer 07 MATH00 Solution to Exercise 5. Find the partial derivatives of the following functions: (a (xy 5z/( + x, (b x/ x + y, (c arctan y/x, (d log((t + 3 + ts, (e sin(xy z 3, (f x α, x = (x,, x n. (a
More informationQualitative behaviour of numerical methods for SDEs and application to homogenization
Qualitative behaviour of numerical methods for SDEs and application to homogenization K. C. Zygalakis Oxford Centre For Collaborative Applied Mathematics, University of Oxford. Center for Nonlinear Analysis,
More informationA Semi-Lagrangian scheme for a regularized version of the Hughes model for pedestrian flow
A Semi-Lagrangian scheme for a regularized version of the Hughes model for pedestrian flow Adriano FESTA Johann Radon Institute for Computational and Applied Mathematics (RICAM) Austrian Academy of Sciences
More informationNANO 243/CENG 207 Course Use Only
L8. Drug Dispersion and Diffusion in Biological Systems (2) April 26, 2018 2. Diffusion in Water (3) Diffusion of small molecule in water drug molecules/solvents are identical to solvent Assume molecules
More informationSuggested Solution to Assignment 7
MATH 422 (25-6) partial diferential equations Suggested Solution to Assignment 7 Exercise 7.. Suppose there exists one non-constant harmonic function u in, which attains its maximum M at x. Then by the
More informationKinetic Theory. Motivation - Relaxation Processes Violent Relaxation Thermodynamics of self-gravitating system
Kinetic Theory Motivation - Relaxation Processes Violent Relaxation Thermodynamics of self-gravitating system negative heat capacity the gravothermal catastrophe The Fokker-Planck approximation Master
More informationarxiv: v1 [q-bio.bm] 20 Dec 2007
Abstract arxiv:0712.3467v1 [q-bio.bm] 20 Dec 2007 The mean time required by a transcription factor (TF) or an enzyme to find a target in the nucleus is of prime importance for the initialization of transcription,
More informationLecture 12: Detailed balance and Eigenfunction methods
Lecture 12: Detailed balance and Eigenfunction methods Readings Recommended: Pavliotis [2014] 4.5-4.7 (eigenfunction methods and reversibility), 4.2-4.4 (explicit examples of eigenfunction methods) Gardiner
More informationBROWNIAN DYNAMICS SIMULATIONS WITH HYDRODYNAMICS. Abstract
BROWNIAN DYNAMICS SIMULATIONS WITH HYDRODYNAMICS Juan J. Cerdà 1 1 Institut für Computerphysik, Pfaffenwaldring 27, Universität Stuttgart, 70569 Stuttgart, Germany. (Dated: July 21, 2009) Abstract - 1
More informationA stochastic particle system for the Burgers equation.
A stochastic particle system for the Burgers equation. Alexei Novikov Department of Mathematics Penn State University with Gautam Iyer (Carnegie Mellon) supported by NSF Burgers equation t u t + u x u
More informationNonlinear Systems and Control Lecture # 19 Perturbed Systems & Input-to-State Stability
p. 1/1 Nonlinear Systems and Control Lecture # 19 Perturbed Systems & Input-to-State Stability p. 2/1 Perturbed Systems: Nonvanishing Perturbation Nominal System: Perturbed System: ẋ = f(x), f(0) = 0 ẋ
More informationMathematical Modelling of Partially Absorbing Boundaries in Biological Systems
Mathematical Modelling of Partially Absorbing Boundaries in Biological Systems by Sarafa Adewale Iyaniwura B.Sc., University of Ilorin, 2012 M.Sc., AIMS-Stellenbosch University, 2014 A THESIS SUBMITTED
More informationPartial Differential Equations
M3M3 Partial Differential Equations Solutions to problem sheet 3/4 1* (i) Show that the second order linear differential operators L and M, defined in some domain Ω R n, and given by Mφ = Lφ = j=1 j=1
More informationLecture 12: Detailed balance and Eigenfunction methods
Miranda Holmes-Cerfon Applied Stochastic Analysis, Spring 2015 Lecture 12: Detailed balance and Eigenfunction methods Readings Recommended: Pavliotis [2014] 4.5-4.7 (eigenfunction methods and reversibility),
More informationModelling nucleocytoplasmic transport with application to the intracellular dynamics of the tumor suppressor protein p53
Modelling nucleocytoplasmic transport with application to the intracellular dynamics of the tumor suppressor protein p53 L. Dimitrio Supervisors: R. Natalini (IAC-CNR) and J. Clairambault (INRIA) 5 Septembre
More informationNonlinear Dynamical Systems Lecture - 01
Nonlinear Dynamical Systems Lecture - 01 Alexandre Nolasco de Carvalho August 08, 2017 Presentation Course contents Aims and purpose of the course Bibliography Motivation To explain what is a dynamical
More informationConnection to Laplacian in spherical coordinates (Chapter 13)
Connection to Laplacian in spherical coordinates (Chapter 13) We might often encounter the Laplace equation and spherical coordinates might be the most convenient 2 u(r, θ, φ) = 0 We already saw in Chapter
More informationPulling forces in Cell Division
Pulling forces in Cell Division Frank Jülicher Max Planck Institute for the Physics of Complex Systems Dresden, Germany Max Planck Institute for the Physics of Complex Systems A. Zumdieck A. J.-Dalmaroni
More informationPartial Differential Equations with Applications to Finance Seminar 1: Proving and applying Dynkin s formula
Partial Differential Equations with Applications to Finance Seminar 1: Proving and applying Dynkin s formula Group 4: Bertan Yilmaz, Richard Oti-Aboagye and Di Liu May, 15 Chapter 1 Proving Dynkin s formula
More informationSimulation methods for stochastic models in chemistry
Simulation methods for stochastic models in chemistry David F. Anderson anderson@math.wisc.edu Department of Mathematics University of Wisconsin - Madison SIAM: Barcelona June 4th, 21 Overview 1. Notation
More informationLangevin Methods. Burkhard Dünweg Max Planck Institute for Polymer Research Ackermannweg 10 D Mainz Germany
Langevin Methods Burkhard Dünweg Max Planck Institute for Polymer Research Ackermannweg 1 D 55128 Mainz Germany Motivation Original idea: Fast and slow degrees of freedom Example: Brownian motion Replace
More informationHomogenization with stochastic differential equations
Homogenization with stochastic differential equations Scott Hottovy shottovy@math.arizona.edu University of Arizona Program in Applied Mathematics October 12, 2011 Modeling with SDE Use SDE to model system
More informationUltra-Cold Plasma: Ion Motion
Ultra-Cold Plasma: Ion Motion F. Robicheaux Physics Department, Auburn University Collaborator: James D. Hanson This work supported by the DOE. Discussion w/ experimentalists: Rolston, Roberts, Killian,
More informationGaussian processes for inference in stochastic differential equations
Gaussian processes for inference in stochastic differential equations Manfred Opper, AI group, TU Berlin November 6, 2017 Manfred Opper, AI group, TU Berlin (TU Berlin) inference in SDE November 6, 2017
More informationInverse Langevin approach to time-series data analysis
Inverse Langevin approach to time-series data analysis Fábio Macêdo Mendes Anníbal Dias Figueiredo Neto Universidade de Brasília Saratoga Springs, MaxEnt 2007 Outline 1 Motivation 2 Outline 1 Motivation
More informationSOLUTION OF THE DIRICHLET PROBLEM WITH A VARIATIONAL METHOD. 1. Dirichlet integral
SOLUTION OF THE DIRICHLET PROBLEM WITH A VARIATIONAL METHOD CRISTIAN E. GUTIÉRREZ FEBRUARY 3, 29. Dirichlet integral Let f C( ) with open and bounded. Let H = {u C ( ) : u = f on } and D(u) = Du(x) 2 dx.
More informationUNIVERSITY OF MANITOBA
Question Points Score INSTRUCTIONS TO STUDENTS: This is a 6 hour examination. No extra time will be given. No texts, notes, or other aids are permitted. There are no calculators, cellphones or electronic
More informationSOLUTIONS TO THE FINAL EXAM. December 14, 2010, 9:00am-12:00 (3 hours)
SOLUTIONS TO THE 18.02 FINAL EXAM BJORN POONEN December 14, 2010, 9:00am-12:00 (3 hours) 1) For each of (a)-(e) below: If the statement is true, write TRUE. If the statement is false, write FALSE. (Please
More informationThe dynamics of small particles whose size is roughly 1 µmt or. smaller, in a fluid at room temperature, is extremely erratic, and is
1 I. BROWNIAN MOTION The dynamics of small particles whose size is roughly 1 µmt or smaller, in a fluid at room temperature, is extremely erratic, and is called Brownian motion. The velocity of such particles
More information5A Effects of receptor clustering and rebinding on intracellular signaling
5A Effects of receptor clustering and rebinding on intracellular signaling In the analysis of bacterial chemotaxis, we showed how receptor clustering can amplify a biochemical signal through cooperative
More informationUncertainty quantification and systemic risk
Uncertainty quantification and systemic risk Josselin Garnier (Université Paris Diderot) with George Papanicolaou and Tzu-Wei Yang (Stanford University) February 3, 2016 Modeling systemic risk We consider
More informationSolution Set of Homework # 2. Friday, September 09, 2017
Temple University Department of Physics Quantum Mechanics II Physics 57 Fall Semester 17 Z. Meziani Quantum Mechanics Textboo Volume II Solution Set of Homewor # Friday, September 9, 17 Problem # 1 In
More informationGeneral Relativity and Compact Objects Neutron Stars and Black Holes
1 General Relativity and Compact Objects Neutron Stars and Black Holes We confine attention to spherically symmetric configurations. The metric for the static case can generally be written ds 2 = e λ(r)
More informationIntroduction to Nonlinear Control Lecture # 3 Time-Varying and Perturbed Systems
p. 1/5 Introduction to Nonlinear Control Lecture # 3 Time-Varying and Perturbed Systems p. 2/5 Time-varying Systems ẋ = f(t, x) f(t, x) is piecewise continuous in t and locally Lipschitz in x for all t
More informationLocal time path integrals and their application to Lévy random walks
Local time path integrals and their application to Lévy random walks Václav Zatloukal (www.zatlovac.eu) Faculty of Nuclear Sciences and Physical Engineering Czech Technical University in Prague talk given
More informationPAPER 311 BLACK HOLES
MATHEMATICAL TRIPOS Part III Friday, 8 June, 018 9:00 am to 1:00 pm PAPER 311 BLACK HOLES Attempt no more than THREE questions. There are FOUR questions in total. The questions carry equal weight. STATIONERY
More informationBridging the Gap between Center and Tail for Multiscale Processes
Bridging the Gap between Center and Tail for Multiscale Processes Matthew R. Morse Department of Mathematics and Statistics Boston University BU-Keio 2016, August 16 Matthew R. Morse (BU) Moderate Deviations
More informationOptimizing the Principal Eigenvalue of the Laplacian in a Sphere with Interior Traps
submitted for the special issue of Mathematical and Computer Modeling based on the international meeting Mathematical Methods and Modeling in the Biological Sciences, held in Rio de Janiero, March 2009
More informationAnalytic solutions of the geodesic equation in static spherically symmetric spacetimes in higher dimensions
Analytic solutions of the geodesic equation in static spherically symmetric spacetimes in higher dimensions Eva Hackmann 2, Valeria Kagramanova, Jutta Kunz, Claus Lämmerzahl 2 Oldenburg University, Germany
More informationPressure and forces in active matter
1 Pressure and forces in active matter Alex Solon (MIT) University of Houston, February 8th 2018 Active matter 2 Stored energy Mechanical energy Self-propelled particles Found at all scales in living systems
More informationModeling of Micro-Fluidics by a Dissipative Particle Dynamics Method. Justyna Czerwinska
Modeling of Micro-Fluidics by a Dissipative Particle Dynamics Method Justyna Czerwinska Scales and Physical Models years Time hours Engineering Design Limit Process Design minutes Continious Mechanics
More informationThe Kramers problem and first passage times.
Chapter 8 The Kramers problem and first passage times. The Kramers problem is to find the rate at which a Brownian particle escapes from a potential well over a potential barrier. One method of attack
More informationTwo dimensional oscillator and central forces
Two dimensional oscillator and central forces September 4, 04 Hooke s law in two dimensions Consider a radial Hooke s law force in -dimensions, F = kr where the force is along the radial unit vector and
More informationPROBLEM SET 6. E [w] = 1 2 D. is the smallest for w = u, where u is the solution of the Neumann problem. u = 0 in D u = h (x) on D,
PROBLEM SET 6 UE ATE: - APR 25 Chap 7, 9. Questions are either directl from the text or a small variation of a problem in the text. Collaboration is oka, but final submission must be written individuall.
More informationGillespie s Algorithm and its Approximations. Des Higham Department of Mathematics and Statistics University of Strathclyde
Gillespie s Algorithm and its Approximations Des Higham Department of Mathematics and Statistics University of Strathclyde djh@maths.strath.ac.uk The Three Lectures 1 Gillespie s algorithm and its relation
More informationExercise sheet 3: Random walk and diffusion equation, the Wiener-Khinchin theorem, correlation function and power spectral density
AMSI (217) Stochastic Equations and Processes in Physics and Biology Exercise sheet 3: Random walk and diffusion equation, the Wiener-Khinchin theorem, correlation function and power spectral density 1.
More informationGene Expression as a Stochastic Process: From Gene Number Distributions to Protein Statistics and Back
Gene Expression as a Stochastic Process: From Gene Number Distributions to Protein Statistics and Back June 19, 2007 Motivation & Basics A Stochastic Approach to Gene Expression Application to Experimental
More informationExit times of diffusions with incompressible drift
Exit times of diffusions with incompressible drift Gautam Iyer, Carnegie Mellon University gautam@math.cmu.edu Collaborators: Alexei Novikov, Penn. State Lenya Ryzhik, Stanford University Andrej Zlatoš,
More informationFirst Passage Time Calculations
First Passage Time Calculations Friday, April 24, 2015 2:01 PM Homework 4 will be posted over the weekend; due Wednesday, May 13 at 5 PM. We'll now develop some framework for calculating properties of
More information08. Brownian Motion. University of Rhode Island. Gerhard Müller University of Rhode Island,
University of Rhode Island DigitalCommons@URI Nonequilibrium Statistical Physics Physics Course Materials 1-19-215 8. Brownian Motion Gerhard Müller University of Rhode Island, gmuller@uri.edu Follow this
More informationPoisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation
Poisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation Jingyi Zhu Department of Mathematics University of Utah zhu@math.utah.edu Collaborator: Marco Avellaneda (Courant
More informationSpotlight on Laplace s Equation
16 Spotlight on Laplace s Equation Reference: Sections 1.1,1.2, and 1.5. Laplace s equation is the undriven, linear, second-order PDE 2 u = (1) We defined diffusivity on page 587. where 2 is the Laplacian
More informationI ml. g l. sin. l 0,,2,...,n A. ( t vs. ), and ( vs. ). d dt. sin l
Advanced Control Theory Homework #3 Student No.597 Name : Jinseong Kim. The simple pendulum dynamics is given by g sin L A. Derive above dynamic equation from the free body diagram. B. Find the equilibrium
More information