Abstract
BackgroundIn cell signaling terminology, adaptation refers to a system's capability of returning to its equilibrium upon a transient response. To achieve this, a network has to be both sensitive and precise. Namely, the system must display a significant output response upon stimulation, and later on return to pre-stimulation levels. If the system settles at the exact same equilibrium, adaptation is said to be 'perfect'. Examples of adaptation mechanisms include temperature regulation, calcium regulation and bacterial chemotaxis.ResultsWe present models of the simplest adaptation architecture, a two-state protein system, in a stochastic setting. Furthermore, we consider differences between individual and collective adaptive behavior, and show how our system displays fold-change detection properties. Our analysis and simulations highlight why adaptation needs to be understood in terms of probability, and not in strict numbers of molecules. Most importantly, selection of appropriate parameters in this simple linear setting may yield populations of cells displaying adaptation, while single cells do not.ConclusionsSingle cell behavior cannot be inferred from population measurements and, sometimes, collective behavior cannot be determined from the individuals. By consequence, adaptation can many times be considered a purely emergent property of the collective system. This is a clear example where biological ergodicity cannot be assumed, just as is also the case when cell replication rates are not homogeneous, or depend on the cell state. Our analysis shows, for the first time, how ergodicity cannot be taken for granted in simple linear examples either. The latter holds even when cells are considered isolated and devoid of replication capabilities (cell-cycle arrested). We also show how a simple linear adaptation scheme displays fold-change detection properties, and how rupture of ergodicity prevails in scenarios where transitions between protein states are mediated by other molecular species in the system, such as phosphatases and kinases.
Highlights
In cell signaling terminology, adaptation refers to a system’s capability of returning to its equilibrium upon a transient response
In this paper, we have studied the effects of stochasticity in a ‘two-state protein’ scheme, providing an explanation of what adaptation means and entails in a stochastic setting
That an adaptation profile can be achieved by calculating the first moment of the chemical master equation (CME), but that the underlying probability distribution might be wide enough to prevent one from making definite quick-anddirty assertions going from a single cell to the population level, or the other way around
Summary
Adaptation refers to a system’s capability of returning to its equilibrium upon a transient response. Despite the intrinsic uncertainty in the occurrence of these chemical events, and basically against all odds, cells prevail as efficient decision makers. Are their fate decisions influenced by stochastic events and embedded within widely fluctuating environments, but they are stochastic themselves [7], the underlying mechanisms of which remain widely unknown. One cannot help but wonder: how do cells process widely varying information from their environment, Adaptive behavior can result from three basic signaling motifs: integral control, negative feedback, and feedforward regulation [8]. A cellular system may proceed in a similar fashion, by comparing ‘actual’ to ‘desired’ conditions, as has been found to be the case in bacterial chemotaxis [10,11,12,13] or calcium homeostasis [14]
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