Transport in a molecular motor system

Chipot, Michel; Hastings, S. P.; Kinderlehrer, David

doi:10.1051/m2an:2004048

Cited by 21 publications

(42 citation statements)

References 28 publications

(31 reference statements)

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“…The term "flashing ratchets" refers to the case of molecules diffusing in a potential with space-time periodic switches and diffusion of size comparable to the period of the potential. Following the setting proposed in [13,14,27,29] (see [20] for another), the molecules are presented by the density n ε (x, t) which evolves by the Fokker-Planck equation ⎧ ⎨ ⎩ n ε,t − εn ε,xx − W y ( x ε , t ε )n ε x = 0 in x ∈ (0, 1) × R, εn ε,x + W y ( x ε , t ε )n ε = 0 at x = 0, 1 and t ∈ R, n ε is ε -periodic in t, n ε > 0, and 1 0 n ε (x, t)dx = 1 for all t ∈ R, (1.1) which is a Floquet eigenproblem that has a unique positive solution (up to multiplication by a constant which explains the unit mass normalization).…”

Section: Introductionmentioning

confidence: 99%

“…Several molecular motor models were analyzed, for fixed ε, in [13,14,20,27,29] using arguments from optimal transportation or ordinary differential equations. The typical results about biomotors obtained in [13,27,33], without periodic potentials, are that, for small diffusion ε and under some precise asymmetry assumptions on the potential and rates, the solutions tend to concentrate, as ε → 0, as Dirac masses at either x = 0 or x = 1.…”

Section: Introductionmentioning

confidence: 99%

“…The typical results about biomotors obtained in [13,27,33], without periodic potentials, are that, for small diffusion ε and under some precise asymmetry assumptions on the potential and rates, the solutions tend to concentrate, as ε → 0, as Dirac masses at either x = 0 or x = 1. This behavior is referred to as "motor effect".…”

Section: Introductionmentioning

confidence: 99%

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A homogenization approach to flashing ratchets

Perthame

Souganidis

2010

Nonlinear Differ. Equ. Appl.

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Abstract. The paper continues the program of the authors to develop a mathematical framework to understand and characterize the notion of "asymmetric" potentials, which has been introduced to explain how molecular motors work, considering flashing ratchets, i.e., molecules diffusing in a potential with periodic switches. The mathematical model is a Fokker-Planck equation with a space-time periodic potential and diffusion of order of magnitude compatible with the period of the potential. After performing a homogenization analysis of the problem the "asymmetric" potentials are characterized by the property that the solution, which models the molecule density, concentrates on one end of the domain. Finally explicit examples are presented exhibiting that the concentration phenomena (motor effect) takes place are presented. The proof uses techniques from the theory of viscosity solutions for the HamiltonJacobi equation which, in the homogenization limit, defines the effective hamiltonian. Mathematics Subject Classification (2000). 35B25, 35B27, 49L25, 92C05.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A homogenization approach to flashing ratchets

Perthame

Souganidis

2010

Nonlinear Differ. Equ. Appl.

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“…A derivation of the system from a mass transport viewpoint is given in [6]. For an analysis of the steady state solutions and for further references we refer to [5], [11], [12], [15] and [16]. In this paper we study the corresponding system in higher space dimension, namely…”

Section: Introductionmentioning

confidence: 99%

Contraction in L^1 for a system arising in chemical reactions and molecular motors

Chipot¹,

Hilhorst²,

Kinderlehrer³

et al. 2009

Differential Equations &Amp; Applications

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Abstract. We prove a contraction in L 1 property for the solutions of a nonlinear reactiondiffusion system whose special cases include a system related to intracellular transport as well as reversible chemical reactions. We then consider the special case of the linear molecular motor problem and prove the existence and uniqueness of the stationary solution up to a multiplicative constant, extending to arbitrary space dimension results which were already known in the one dimensional case; this in turn implies the convergence to stationary solutions of the solutions of the time evolution linear molecular motor problem.

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“…There are two ways to attack this, one starting with the Schauder Fixed Point Theorem and one by a shooting method, based on writing (17), (18) as a first order system, [4]. We would like to briefly discuss the origins of transport and the role of the asymmetry of the potentials.…”

Section: The Stationary Solutionmentioning

confidence: 99%

Remarks About Diffusion Mediated Transport: Thinking About Motion in Small Systems

Hastings

Kinderlehrer

Variational Analysis and Applications

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We describe a dissipation principle/variational principle which may be useful in modeling motion in small viscous systems and provide brief illustrations to brownian motor or molecular rachet situations which are found in intracellular transport. Monge-Kantorovich mass transport and Wasserstein metric play an interesting role in these developments. Some properties of the system that ensure the presence of transport are discussed.

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Transport in a molecular motor system

Cited by 21 publications

References 28 publications

A homogenization approach to flashing ratchets

A homogenization approach to flashing ratchets

Contraction in L^1 for a system arising in chemical reactions and molecular motors

Remarks About Diffusion Mediated Transport: Thinking About Motion in Small Systems

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