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Models at the single cell level

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Many cellular behaviors cannot be completely captured or appropriately described at the cell population level. Noise induced by stochastic chemical reactions, spatially polarized signaling networks, and heterogeneous cell–cell communication are among the many phenomena that require fine‐grained analysis. Accordingly, the mathematical models used to describe such systems must be capable of single cell or subcellular resolution. Here, we review techniques for modeling single cells, including models of stochastic chemical kinetics, spatially heterogeneous intracellular signaling, and spatial stochastic systems. We also briefly discuss applications of each type of model. Copyright © 2009 John Wiley & Sons, Inc.

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Figure 1.

Examples of single cell behaviors that produce the same average population behavior. Individual cells are shown as circles with each cell's signaling activity denoted by its color; black denotes 100% signaling and white denotes 0% signaling. The population average for each panel is 50%. In panel (a), each cell signals at 50%. In panel (b), half of the cells are in the 100% state and half of the cells are in the 0% state. In panel (c), cells are as in panel (b), but individual cells may switch states in time without affecting the population average. The switching may occur stochastically (panel (c)), or result from oscillatory signaling activity whose phase and frequency vary from cell to cell (panel (d)). In panel (e), cell signaling is distributed between 0% and 100%. In panel (f), signaling within a cell is spatially heterogeneous such that the average signaling within each cell is 50%.

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Figure 2.

Schematic showing the Gillespie algorithm and its variations. Panels (a) and (b) show examples of the Gillespie algorithm applied to the λ phage system, as described in the text (reproduced with permission from Ref 44. Copyright 1997). Panels (c)–(e) show schematics of different implementations of the Gillespie algorithm. In this schematic, the chemical system is composed of 3 different reactions. In each diagram, the axis represents time, tick marks represent iterations of the algorithm, and the labels represent the reaction(s) that occur during each iteration. Panel (c) illustrates the direct method in which one reaction (rxn) occurs per iteration. Panel (d) illustrates the tau‐leaping method, in which multiple instances of multiple reactions may occur per iteration. Panel (e) illustrates a hybrid method, in which one instance of the slow reaction (reaction 3) occurs per iteration, while the other reactions are updated by some other technique. In each method, the period of time corresponding to each iteration is chosen stochastically, with tau‐leaping and hybrid methods using generally larger time steps than the direct method.

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Figure 3.

Pattern generation through Turing‐like mechanisms. The Turing mechanism shows that a two‐component system can generate a spatially heterogeneous pattern when the diffusion coefficients are significantly different, as shown in (a). The two basic mechanisms of pattern formation in a two‐component system are the activator‐inhibitor (AI) mechanism (b) and the substrate‐depletion (SD) mechanism (c), as described in the text. Both panels show the connection diagrams for the respective mechanism above and a possible biological implementation below. (d) The yeast bud‐site selection and polarization mechanism offers a unique example of parallel Turing mechanisms in biology. As shown in panel (i), the WT GTPase Cdc42 displays a polarized response corresponding to the presumptive bud site, as do the two mutant versions of Cdcd42 i.e. Q61L [constitutively GTP bound form] and D57Y (constitutively GDP bound form).91 The membrane bound polarized profile of a cell with the Cdc42‐Q61L mutation (ii) is plotted in (iii).80 The formation of the incipient bud can be divided into two parallel phases (iv). The initial symmetry‐breaking phase involves the formation of a cluster of activated Cdc42, which can occur without the presence of actin or microtubule machinery, and requires a positive feedback mechanism involving Cdc42, its GEF Cdc24 and the scaffold protein Bem1 (which binds both Cdc24 and Cdc42GTP). In particular, this polarization depends on the cycling of Cdc42 between the GDP and GTP bound forms. The second phase involves an actin‐mediated positive feedback involving polarized secretion of vesicles (containing proteins like Cdc42) along actin cables leads to an ensuing phase of growth and protrusion. This mechanism does not depend on Bem1 and Cdc24 and leads to the polarized distribution of the Q61L and D57Y mutants. Mathematical models of each of these phases display polarization through inherently Turing‐based mechanisms. Panel (v) shows a model of the initial actin‐independent phase which leads to a polarized Cdc42 distribution as shown in (vi).92 Mathematical modeling shows that a combination of lateral diffusion of Cdc42, its endocytosis and its polarized secretion as part of a positive feedback (increased presence of Cdc42 leads to increased probability of actin cable formation and decreased actin cable detachment) as shown in panel (vii) can lead to a polarized distribution of Cdc42 as shown in panel (viii).80 WT, wildtype; GDP, guanosine diphosphate; GTP, guanosine triphosphate; GEF, guanine nucleotide exchange factor. (Panels (i) and (iv) reprinted with permission from Ref 91. Copyright 2004 The Rockefeller University Press). (Panels (ii), (iii), (vii), (viii) reprinted with permission from Ref 80. Copyright 2007 Elsevier). (Panels (v) and (vi) reprinted with permission from Ref 92. Copyright 2008 FEBS)

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Figure 4.

Example of spatial stochastic modeling. Panel (a) shows a schematic of an M‐Cell model used to model cardiac myocyte dyadic clefts (the narrow space between the cell membrane and sarcoplasmic reticulum). Calcium ions are shown in red, L‐type calcium channels in blue, and ryanodine receptors in yellow. Panel (b) shows an example of an M‐Cell simulation showing calcium signaling in the dyadic cleft. M‐Cell tracks the location, random movement, and identity of each molecule within the system volume, thereby providing a spatial stochastic simulation. (Reprinted with permission from Ref 124. Copyright 2006 Biophysical Society.).

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Jens Nielsen

Jens Nielsen
is a Professor in the Department of Biology and Biological Engineering at Chalmers University of Technology in Göteborg, Sweden. His research focus is on systems biology of metabolism. The yeast Saccharomyces cerevisiae is the lab’s key organism for experimental research, but they also work with Aspergilli oryzae.

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