اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدRigorous bounds on the stationary distributions of the chemical master equation via mathematical programming
164
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
The stochastic dynamics of biochemical networks are usually modelled with the chemical master equation (CME). The stationary distributions of CMEs are seldom solvable analytically, and numerical methods typically produce estimates with uncontrolled errors. Here, we introduce mathematical programming approaches that yield approximations of these distributions with computable error bounds which enable the verification of their accuracy. First, we use semidefinite programming to compute increasingly tighter upper and lower bounds on the moments of the stationary distributions for networks with rational propensities. Second, we use these moment bounds to formulate linear programs that yield convergent upper and lower bounds on the stationary distributions themselves, their marginals and stationary averages. The bounds obtained also provide a computational test for the uniqueness of the distribution. In the unique case, the bounds form an approximation of the stationary distribution with a computable bound on its error. In the non-unique case, our approach yields converging approximations of the ergodic distributions. We illustrate our methodology through several biochemical examples taken from the literature: Schlogls model for a chemical bifurcation, a two-dimensional toggle switch, a model for bursty gene expression, and a dimerisation model with multiple stationary distributions.
قيم البحث
اقرأ أيضاً
We consider the relationship between stationary distributions for stochastic models of reaction systems and Lyapunov functions for their deterministic counterparts. Specifically, we derive the well known Lyapunov function of reaction network theory a
s a scaling limit of the non-equilibrium potential of the stationary distribution of stochastically modeled complex balanced systems. We extend this result to general birth-death models and demonstrate via example that similar scaling limits can yield Lyapunov functions even for models that are not complex or detailed balanced, and may even have multiple equilibria.
Master equations are common descriptions of mesoscopic systems. Analytical solutions to these equations can rarely be obtained. We here derive an analytical approximation of the time-dependent probability distribution of the master equation using ort
hogonal polynomials. The solution is given in two alternative formulations: a series with continuous and a series with discrete support both of which can be systematically truncated. While both approximations satisfy the system size expansion of the master equation, the continuous distribution approximations become increasingly negative and tend to oscillations with increasing truncation order. In contrast, the discrete approximations rapidly converge to the underlying non-Gaussian distributions. The theory is shown to lead to particularly simple analytical expressions for the probability distributions of molecule numbers in metabolic reactions and gene expression systems.
We consider stochastically modeled chemical reaction systems with mass-action kinetics and prove that a product-form stationary distribution exists for each closed, irreducible subset of the state space if an analogous deterministically modeled syste
m with mass-action kinetics admits a complex balanced equilibrium. Feinbergs deficiency zero theorem then implies that such a distribution exists so long as the corresponding chemical network is weakly reversible and has a deficiency of zero. The main parameter of the stationary distribution for the stochastically modeled system is a complex balanced equilibrium value for the corresponding deterministically modeled system. We also generalize our main result to some non-mass-action kinetics.
Network growth as described by the Duplication-Divergence model proposes a simple general idea for the evolution dynamics of natural networks. In particular it is an alternative to the well known Barabasi-Albert model when applied to protein-protein
interaction networks. In this work we derive a master equation for the node degree distribution of networks growing via Duplication and Divergence and we obtain an expression for the total number of links and for the degree distribution as a function of the number of nodes. Using algebra tools we investigate the degree distribution asymptotic behavior. Analytic results show that the network nodes average degree converges if the total mutation rate is greater than 0.5 and diverges otherwise. Treating original and duplicated node mutation rates as independent parameters has no effect on this result. However, difference in these parameters results in a slower rate of convergence and in different degree distributions. The more different these parameters are, the denser the tail of the distribution. We compare the solutions obtained with simulated networks. These results are in good agreement with the expected values from the derived expressions. The method developed is a robust tool to investigate other models for network growing dynamics.
State space truncation methods are widely used to approximate solutions of the chemical master equation. While most methods of this kind focus on truncating the state space directly, in this work we propose modifying the underlying chemical reaction
network by introducing emph{slack reactants} that indirectly truncate the state space. More specifically, slack reactants introduce an expanded chemical reaction network and impose a truncation scheme based on user defined properties, such as mass-conservation. This network structure allows us to prove inheritance of special properties of the original model, such as irreducibility and complex balancing. We use the network structure imposed by slack reactants to prove the convergence of the stationary distribution and first arrival times. We then provide examples comparing our method with the stationary finite state projection and finite buffer methods. Our slack reactant system appears to be more robust than some competing methods with respect to calculating first arrival times.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
