Abstract
Biochemical processes are inherently stochastic, creating molecular fluctuations in otherwise identical cells. Such "noise" is widespread but has proven difficult to analyze because most systems are sparsely characterized at the single cell level and because nonlinear stochastic models are analytically intractable. Here, we exactly relate average abundances, lifetimes, step sizes, and covariances for any pair of components in complex stochastic reaction systems even when the dynamics of other components are left unspecified. Using basic mathematical inequalities, we then establish bounds for whole classes of systems. These bounds highlight fundamental trade-offs that show how efficient assembly processes must invariably exhibit large fluctuations in subunit levels and how eliminating fluctuations in one cellular component requires creating heterogeneity in another.
Highlights
This may seem impossible, and it is if the goal is to obtain closed-form expressions capturing a system’s behavior: most nonlinear stochastic models are analytically intractable, and the question of how a system behaves is not even well posed unless all parts are specified
Biochemical processes are inherently stochastic, creating molecular fluctuations in otherwise identical cells. Such “noise” is widespread but has proven difficult to analyze because most systems are sparsely characterized at the single cell level and because nonlinear stochastic models are analytically intractable
To be broadly applicable in biology, such generalized analytical approaches would need to account for inherently stochastic processes far from thermodynamic equilibrium and allow for nonlinear reaction rates of adding or removing individual components in discrete steps or bursts—without linearizations or Gaussian approximations. They should be formulated in terms of properties that have clear physical definitions or can be estimated experimentally, and—most importantly—be able to make strong statements about sparsely characterized reaction networks without ignoring or guessing the unknown parts
Summary
Biochemical processes are inherently stochastic, creating molecular fluctuations in otherwise identical cells. To be broadly applicable in biology, such generalized analytical approaches would need to account for inherently stochastic processes far from thermodynamic equilibrium and allow for nonlinear reaction rates of adding or removing individual components in discrete steps or bursts—without linearizations or Gaussian approximations They should be formulated in terms of properties that have clear physical definitions or can be estimated experimentally, and—most importantly—be able to make strong statements about sparsely characterized reaction networks without ignoring or guessing the unknown parts. For averages and variances only directly depend on how the corresponding component is made and degraded Those equations can be combined with basic statistical inequalities to derive general bounds, which, in turn, can be exactly expressed in terms of physical observables that can be experimentally identified without knowing the microscopic details of the system. The average abundances hxii and the (co)variance matrix C with entries Cij 1⁄4 hxixji − hxiihxji are described by
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