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Bridging the gap between stochastic and deterministic regimes in the kinetic simulations of the biochemical reaction networks.

ABSTRACT: The biochemical reaction networks include elementary reactions differing by many orders of magnitude in the numbers of molecules involved. The kinetics of reactions involving small numbers of molecules can be studied by exact stochastic simulation. This approach is not practical for the simulation of metabolic processes because of the computational cost of accounting for individual molecular collisions. We present the "maximal time step method," a novel approach combining the Gibson and Bruck algorithm with the Gillespie tau-leap method. This algorithm allows stochastic simulation of systems composed of both intensive metabolic reactions and regulatory processes involving small numbers of molecules. The method is applied to the simulation of glucose, lactose, and glycerol metabolism in Escherichia coli. The gene expression, signal transduction, transport, and enzymatic activities are modeled simultaneously. We show that random fluctuations in gene expression can propagate to the level of metabolic processes. In the cells switching from glucose to a mixture of lactose and glycerol, random delays in transcription initiation determine whether lactose or glycerol operon is induced. In a small fraction of cells severe decrease in metabolic activity may also occur. Both effects are epigenetically inherited by the progeny of the cell in which the random delay in transcription initiation occurred.

SUBMITTER: Puchalka J

PROVIDER: S-EPMC1303974 | biostudies-literature |

REPOSITORIES: biostudies-literature

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Project description:In the mathematical modeling of biochemical reactions, a convenient standard approach is to use ordinary differential equations (ODEs) that follow the law of mass action. However, this deterministic ansatz is based on simplifications; in particular, it neglects noise, which is inherent to biological processes. In contrast, the stochasticity of reactions is captured in detail by the discrete chemical master equation (CME). Therefore, the CME is frequently applied to mesoscopic systems, where copy numbers of involved components are small and random fluctuations are thus significant. Here, we compare those two common modeling approaches, aiming at identifying parallels and discrepancies between deterministic variables and possible stochastic counterparts like the mean or modes of the state space probability distribution. To that end, a mathematically flexible reaction scheme of autoregulatory gene expression is translated into the corresponding ODE and CME formulations. We show that in the thermodynamic limit, deterministic stable fixed points usually correspond well to the modes in the stationary probability distribution. However, this connection might be disrupted in small systems. The discrepancies are characterized and systematically traced back to the magnitude of the stoichiometric coefficients and to the presence of nonlinear reactions. These factors are found to synergistically promote large and highly asymmetric fluctuations. As a consequence, bistable but unimodal, and monostable but bimodal systems can emerge. This clearly challenges the role of ODE modeling in the description of cellular signaling and regulation, where some of the involved components usually occur in low copy numbers. Nevertheless, systems whose bimodality originates from deterministic bistability are found to sustain a more robust separation of the two states compared to bimodal, but monostable systems. In regulatory circuits that require precise coordination, ODE modeling is thus still expected to provide relevant indications on the underlying dynamics.

Project description:The condensational growth of submicrometer aerosol particles to climate relevant sizes is sensitive to their ability to accommodate vapor molecules, which is described by the mass accommodation coefficient. However, the underlying processes are not yet fully understood. We have simulated the mass accommodation and evaporation processes of water using molecular dynamics, and the results are compared to the condensation equations derived from the kinetic gas theory to shed light on the compatibility of the two. Molecular dynamics simulations were performed for a planar TIP4P-Ew water surface at four temperatures in the range 268-300 K as well as two droplets, with radii of 1.92 and 4.14 nm at T = 273.15 K. The evaporation flux from molecular dynamics was found to be in good qualitative agreement with that predicted by the simple kinetic condensation equations. Water droplet growth was also modeled with the kinetic multilayer model KM-GAP of Shiraiwa et al. [Atmos. Chem. Phys. 2012, 12, 2777]. It was found that, due to the fast transport across the interface, the growth of a pure water droplet is controlled by gas phase diffusion. These facts indicate that the simple kinetic treatment is sufficient in describing pure water condensation and evaporation. The droplet size was found to have minimal effect on the value of the mass accommodation coefficient. The mass accommodation coefficient was found to be unity (within 0.004) for all studied surfaces, which is in agreement with previous simulation work. Additionally, the simulated evaporation fluxes imply that the evaporation coefficient is also unity. Comparing the evaporation rates of the mass accommodation and evaporation simulations indicated that the high collision flux, corresponding to high supersaturation, present in typical molecular dynamics mass accommodation simulations can under certain conditions lead to an increase in the evaporation rate. Consequently, in such situations the mass accommodation coefficient can be overestimated, but in the present cases the corrected values were still close to unity with the lowest value at ≈0.99.

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Bridging the gap between stochastic and deterministic regimes in the kinetic simulations of the biochemical reaction networks.

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