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Zamora-Chimal C.,CINVESTAV | Santillan M.,CINVESTAV | Santillan M.,Center for Applied Mathematics in Bioscience and Medicine | Rodriguez-Gonzalez J.,CINVESTAV
Journal of Theoretical Biology | Year: 2012

In this paper we introduce a mathematical model for the tryptophan operon regulatory pathway in Bacillus subtilis. This model considers the transcription-attenuation, and the enzyme-inhibition regulatory mechanisms. Special attention is paid to the estimation of all the model parameters from reported experimental data. With the aid of this model we investigate, from a mathematical-modeling point of view, whether the existing multiplicity of regulatory feedback loops is advantageous in some sense, regarding the dynamic response and the biochemical noise in the system. The tryptophan operon dynamic behavior is studied by means of deterministic numeric simulations, while the biochemical noise is analyzed with the aid of stochastic simulations. The model feasibility is tested comparing its stochastic and deterministic results with experimental reports. Our results for the wildtype and for a couple of mutant bacterial strains suggest that the enzyme-inhibition feedback loop, dynamically accelerates the operon response, and plays a major role in the reduction of biochemical noise. Also, the transcription-attenuation feedback loop makes the trp operon sensitive to changes in the endogenous tryptophan level, and increases the amplitude of the biochemical noise. © 2012 Elsevier Ltd.

Zeron E.S.,National Polytechnic Institute of Mexico | Santillan M.,CINVESTAV | Santillan M.,Center for Applied Mathematics in Bioscience and Medicine
Journal of Theoretical Biology | Year: 2010

In this work we introduce a novel approach to study biochemical noise. It comprises a simplification of the master equation of complex reaction schemes (via an adiabatic approximation) and the numerical solution of the reduced master equation. The accuracy of this procedure is tested by comparing its results with analytic solutions (when available) and with Gillespie stochastic simulations. We further employ our approach to study the stochastic expression of a simple gene network, which is subject to negative feedback regulation at the transcriptional level. Special attention is paid to the influence of negative feedback on the amplitude of intrinsic noise, as well as on the relaxation rate of the system probability distribution function to the steady solution. Our results suggest the existence of an optimal feedback strength that maximizes this relaxation rate. © 2010 Elsevier Ltd.

Zeron E.S.,National Polytechnic Institute of Mexico | Santillan M.,CINVESTAV | Santillan M.,Center for Applied Mathematics in Bioscience and Medicine
Methods in Enzymology | Year: 2011

In this work, we introduce a couple of algorithms to compute the stationary probability distribution for the chemical master equation (CME) of arbitrary chemical networks. We further find the conditions that guaranty the algorithms' convergence and the unicity and stability of the stationary distribution. Next, we employ these algorithms to study the mRNA and protein probability distributions in a gene regulatory network subject to negative feedback regulation. In particular, we analyze the influence of the promoter activation/deactivation speed on the shape of such distributions. We find that a reduction of the promoter activation/deactivation speed modifies the shape of those distributions in a way consistent with the phenomenon known as mRNA (or transcription) bursting. © 2011 Elsevier Inc.

Hilbert L.,McGill University | Hilbert L.,Center for Applied Mathematics in Bioscience and Medicine | Bates G.,McGill University | Roman H.N.,McGill University | And 6 more authors.
PLoS Computational Biology | Year: 2013

The proteins involved in smooth muscle's molecular contractile mechanism - the anti-parallel motion of actin and myosin filaments driven by myosin heads interacting with actin - are found as different isoforms. While their expression levels are altered in disease states, their relevance to the mechanical interaction of myosin with actin is not sufficiently understood. Here, we analyzed in vitro actin filament propulsion by smooth muscle myosin for α-actin (αA), α-actin-tropomyosin-αβ (αA-Tmαβ), α-actin-tropomyosin-β (αA-Tmβ), γ-actin (γA), γ-actin-tropomyosin-αβ (γA-Tmαβ), and γ-actin-tropomoysin-β (γA-Tmβ). Actin sliding analysis with our specifically developed video analysis software followed by statistical assessment (Bootstrapped Principal Component Analysis) indicated that the in vitro motility of αA, γA, and γA-Tmαβ is not distinguishable. Compared to these three 'baseline conditions', statistically significant differences (p<0.05) were: αA-Tmαβ - actin sliding velocity increased 1.12-fold, γA-Tmβ - motile fraction decreased to 0.96-fold, stop time elevated 1.6-fold, αA-Tmβ - run time elevated 1.7-fold. We constructed a mathematical model, simulated actin sliding data, and adjusted the kinetic parameters so as to mimic the experimentally observed differences: αA-Tmαβ - myosin binding to actin, the main, and the secondary myosin power stroke are accelerated, γA-Tmβ - mechanical coupling between myosins is stronger, αA-Tmβ - the secondary power stroke is decelerated and mechanical coupling between myosins is weaker. In summary, our results explain the different regulatory effects that specific combinations of actin and smooth muscle tropomyosin have on smooth muscle actin-myosin interaction kinetics. © 2013 Hilbert et al.

Hilbert L.,McGill University | Hilbert L.,Center for Applied Mathematics in Bioscience and Medicine | Hilbert L.,Meakins Christie Laboratories | Balassy Z.,McGill University | And 6 more authors.
Biophysical Journal | Year: 2015

Actin filaments propelled in vitro by groups of skeletal muscle myosin motors exhibit distinct phases of active sliding or arrest, whose occurrence depends on actin length (L) within a range of up to 1.0 μm. Smooth muscle myosin filaments are exponentially distributed with ≈150 nm average length in vivo - suggesting relevance of the L-dependence of myosin group kinetics. Here, we found L-dependent actin arrest and sliding in in vitro motility assays of smooth muscle myosin. We perturbed individual myosin kinetics with varying, physiological concentrations of phosphate (Pi, release associated with main power stroke) and adenosine diphosphate (ADP, release associated with minor mechanical step). Adenosine triphosphate was kept constant at physiological concentration. Increasing [Pi] lowered the fraction of time for which actin was actively sliding, reflected in reduced average sliding velocity (ν) and motile fraction (fmot, fraction of time that filaments are moving); increasing [ADP] increased the fraction of time actively sliding and reduced the velocity while sliding, reflected in reduced ν and increased fmot. We introduced specific Pi and ADP effects on individual myosin kinetics into our recently developed mathematical model of actin propulsion by myosin groups. Simulations matched our experimental observations and described the inhibition of myosin group kinetics. At low [Pi] and [ADP], actin arrest and sliding were reflected by two distinct chemical states of the myosin group. Upon [Pi] increase, the probability of the active state decreased; upon [ADP] increase, the probability of the active state increased, but the active state became increasingly similar to the arrested state. © 2015 Biophysical Society.

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