Gene regulatory networks: A coarse-grained, equation-free approach to multiscale computation

Erban, Radek; Kevrekidis, Ioannis G.; Adalsteinsson, David; Elston, Timothy C.

doi:10.1063/1.2149854

Cited by 78 publications

(104 citation statements)

References 36 publications

(85 reference statements)

Supporting

Mentioning

101

Contrasting

Unclassified

Order By: Relevance

“…A natural question is what can be done if a suitable projection P is not easily computed, or if the explicit form of G is impossible to obtain because of the complexity of the mapping G. In some cases it may still be possible to describe the macroscopic-level dynamics by the evolution of a few slow variables, and by using computational equation-free methods which are currently being developed, to obtain populational level quantities without explicitly deriving the macroscopic equations (see [29,16,15] and references there), using either the full model of the amoeboid cell or the best available reduction of it.…”

Section: Amoeboid Taxis With Internal Variablesmentioning

confidence: 99%

Taxis equations for amoeboid cells

Erban

Othmer

2007

J. Math. Biol.

Self Cite

View full text Add to dashboard Cite

The classical macroscopic chemotaxis equations have previously been derived from an individual-based description of the tactic response of cells that use a "run-and-tumble" strategy in response to environmental cues [17,18]. Here we derive macroscopic equations for the more complex type of behavioral response characteristic of crawling cells, which detect a signal, extract directional information from a scalar concentration field, and change their motile behavior accordingly. We present several models of increasing complexity for which the derivation of population-level equations is possible, and we show how experimentallymeasured statistics can be obtained from the transport equation formalism. We also show that amoeboid cells that do not adapt to constant signals can still aggregate in steady gradients, but not in response to periodic waves. This is in contrast to the case of cells that use a "run-and-tumble" strategy, where adaptation is essential.

show abstract

Section: Amoeboid Taxis With Internal Variablesmentioning

confidence: 99%

Taxis equations for amoeboid cells

Erban

Othmer

2007

J. Math. Biol.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Consider the reversible chemical reaction [a dimerization, which is a part of several biochemical mechanisms (19,20)] involving two molecular species X and Y ,…”

Section: Multiscale Chemical Reactions: a Toy Examplementioning

confidence: 99%

Detecting intrinsic slow variables in stochastic dynamical systems by anisotropic diffusion maps

Singer

Erban²,

Kevrekidis³

et al. 2009

Proc. Natl. Acad. Sci. U.S.A.

Self Cite

143

182

View full text Add to dashboard Cite

Nonlinear independent component analysis is combined with diffusion-map data analysis techniques to detect good observables in high-dimensional dynamic data. These detections are achieved by integrating local principal component analysis of simulation bursts by using eigenvectors of a Markov matrix describing anisotropic diffusion. The widely applicable procedure, a crucial step in model reduction approaches, is illustrated on stochastic chemical reaction network simulations.slow manifold | dimensionality reduction | chemical reactions E volution of dynamical systems often occurs on two or more time scales. A simple deterministic example is given by the coupled system of ordinary differential equations (ODEs)with the small parameter 0 < τ 1, where α(u, v) and β(u, v) are O(1). For any given initial condition (u 0 , v 0 ), already at t = O(τ) the system approaches a new value (u 0 , v), where v satisfies the asymptotic relation β(u 0 , v) = 0. Although the system is fully described by two coordinates, the relation β(u, v) = 0 defines a slow one-dimensional manifold which approximates the slow dynamics for t τ. In this example, it is clear that v is the fast variable whereas u is the slow one. Projecting onto the slow manifold here is rather easy: The fast foliation is simply "vertical", i.e. u = const. However, when we observe the system in terms of the variables x = x(u, v) and y = y (u, v) which are unknown nonlinear functions of u and v, then the "observables" x and y have both fast and slow dynamics. Projecting onto the slow manifold becomes nontrivial, because the transformation from (x, y) to (u, v) is unknown. Detecting the existence of an intrinsic slow manifold under these conditions and projecting onto it are important in any model reduction technique. Knowledge of a good parametrization of such a slow manifold is a crucial component of the equation-free framework for modeling and computation of complex/multiscale systems (1-3).Principal component analysis (PCA, also known as POD) (4-6) has traditionally been used for data and model reduction in contexts ranging from meteorology (7) and transitional flows (8) to protein folding (9, 10); in these contexts the PCA procedure is used to detect good global reduced coordinates that best capture the data variability. In recent years, diffusion maps (11-17) have been used in a similar spirit to detect low-dimensional, nonlinear manifolds underlying high-dimensional datasets.In this paper, we integrate ensembles of local PCA analyses in the diffusion-map framework to enable the detection of slow variables in high-dimensional data arising from dynamic model simulations. The proposed algorithm is built along the lines of the nonlinear independent component analysis method recently introduced in ref. 18. The approach takes into account the time dependence of the data, whereas in the diffusion-map approach the time labeling of the data points is not included. We demonstrate our algorithm for stochastic simulators arising in the context of chemical/biochemical react...

show abstract

“…Coarse-grained analysis [4] allows us to obtain an effective FPE describing the collective behavior of the locusts at the macroscopic level. By using this coarse-graining technique (see [3]) we were able to extract the coefficients of the assumed underlying FPE describing the alignment of the locusts from the experimental data presented in [1].…”

Section: The Modelmentioning

confidence: 99%

“…The key point in the analysis performed in [3] was the estimation of coefficients of an effective Fokker-Planck equation (FPE) [4], which is written in terms of a macroscopic (low-dimensional) observable [5], the average velocity of the marching group, derived directly from the experimental data. In the present work we approximate the drift and diffusion coefficients of the effective FPE by analytical functions.…”

Section: Introductionmentioning

confidence: 99%

Ergodic directional switching in mobile insect groups

et al. 2010

Self Cite

View full text Add to dashboard Cite

We obtain a Fokker-Planck equation describing experimental data on the collective motion of locusts. The noise is of internal origin and due to the discrete character and finite number of constituents of the swarm. The stationary probability distribution shows a rich phenomenology including non-monotonic behavior of several order/disorder transition indicators in noise intensity.This complex behavior arises naturally as a result of the randomness in the system. Its counterintuitive character challenges standard interpretations of noise induced transitions and calls for an extension of this theory in order to capture the behavior of certain classes of biologically motivated models. Our results suggest that the collective switches of the group's direction of motion might be due to a random ergodic effect and, as such, they are inherent to group formation.

show abstract

Gene regulatory networks: A coarse-grained, equation-free approach to multiscale computation

Cited by 78 publications

References 36 publications

Taxis equations for amoeboid cells

Taxis equations for amoeboid cells

Detecting intrinsic slow variables in stochastic dynamical systems by anisotropic diffusion maps

Ergodic directional switching in mobile insect groups

Contact Info

Product

Resources

About