In order to survive, reproduce and (in multicellular organisms) differentiate, cells must control the concentrations of the myriad different proteins that are encoded in the genome. The precision of this control is limited by the inevitable randomness of individual molecular events. Here we explore how cells can maximize their control power in the presence of these physical limits; formally, we solve the theoretical problem of maximizing the information transferred from inputs to outputs when the number of available molecules is held fixed. We start with the simplest version of the problem, in which a single transcription factor protein controls the readout of one or more genes by binding to DNA. We further simplify by assuming that this regulatory network operates in steady state, that the noise is small relative to the available dynamic range, and that the target genes do not interact. Even in this simple limit, we find a surprisingly rich set of optimal solutions. Importantly, for each locally optimal regulatory network, all parameters are determined once the physical constraints on the number of available molecules are specified. Although we are solving an over--simplified version of the problem facing real cells, we see parallels between the structure of these optimal solutions and the behavior of actual genetic regulatory networks. Subsequent papers will discuss more complete versions of the problem.
Deep Dive into Optimizing information flow in small genetic networks. I.
In order to survive, reproduce and (in multicellular organisms) differentiate, cells must control the concentrations of the myriad different proteins that are encoded in the genome. The precision of this control is limited by the inevitable randomness of individual molecular events. Here we explore how cells can maximize their control power in the presence of these physical limits; formally, we solve the theoretical problem of maximizing the information transferred from inputs to outputs when the number of available molecules is held fixed. We start with the simplest version of the problem, in which a single transcription factor protein controls the readout of one or more genes by binding to DNA. We further simplify by assuming that this regulatory network operates in steady state, that the noise is small relative to the available dynamic range, and that the target genes do not interact. Even in this simple limit, we find a surprisingly rich set of optimal solutions. Importantly, for e
Much of the everyday business of organisms involves the transmission and processing of information. On our human scale, the familiar examples involve the signals taken in through our sense organs [1]. On a cellular scale, information flows from receptors on the cell surface into the cell, modulating biochemical events and ultimately controlling gene expression [2]. In the course of development in multicellular organisms, individual cells acquire information about their location in the embryo by responding to particular "morphogen" molecules whose concentration varies along the main axes of the embryo [3,4]. In all these examples, information of interest to the organism ultimately is represented by events at the molecular level, whether the molecules are transcription factors regulating gene expression or ion channels controlling electrical signals in the brain. This representation is limited by fundamental physical principles: individual molecular events are stochastic, so that with any finite number of molecules there is a limit to the precision with which small signals can be discriminated reliably, and there is a limit to the overall dynamic range of the signals. Our goal in this paper (and its sequel) is to explore these limits to information transmission in the context of small genetic control circuits.
The outputs of genetic control circuits are protein molecules that are synthesized by the cell from messenger RNA (mRNA), which in turn is transcribed from the DNA template. The inputs often are protein molecules as well, “transcription factors” that bind to the DNA and regulate the synthesis of the mRNA. In the last decade, a number of experiments have mapped the input/output relations of these regulatory elements, and characterized their noise, that is the fluctuations in the output protein concentration when the inputs are held fixed [5,6,7,8,9,10,11,12,13,14,15,16,17]. In parallel, a number of theoretical papers have tried to understand the origins of this noise, which ultimately reflects the random behavior of individual molecules along the path from input to output-the arrival of transcription factors at the their targets along the DNA, the initiation of transcription and the degradation of mRNA, the initiation of protein synthesis and the degradation of the output proteins [18,19,20,21,22,23,24,25,26,27,28,29]. While open questions remain, it seems fair to say that we have a physical picture of the noise in genetic control that we can use to ask questions about the overall function and design of these systems.
The ability of any system to transmit information is determined not just by input/output relations and noise levels, but also by the distribution of inputs; maximal information transmission requires a matching between the intrinsic properties of the system and the input statistics [30,31]. In the context of sensory information processing, these matching conditions have been explored almost since the inception of information theory [32,33,34,35]. In particular, because the distribution of sensory inputs varies with time, optimal information transmission requires that the input/output relation track or adapt to these variations, and this theoretical prediction has led to a much richer view of adaptation in the neural code [36,37,38,39,40]. There are analogous matching conditions for genetic regulatory elements, and these condi-tions provide parameter free predictions about the behavior of the system, based on the idea that cells are trying to transmit the maximum amount of information [41]. Comparison with recent experiments has been encouraging [42].
In this paper we go beyond the matching conditions to ask how cells can adjust the input/output relations of genetic regulatory elements so as to maximize the information that is transmitted through these systems. Absent any constraints, the answer will always be to make more molecules, since this reduces the effective noise level, so we consider the problem of maximizing information transmission with a fixed mean or maximum number of molecules at both the input and the output. In this sense we are asking how cells can extract the maximum control power, measured in bits, from a given number of molecules, thus optimizing functionality under clear physical constraints. In general this problem is very difficult, so we start here with the simplest case of a single input transcription factor that controls (potentially) many genes, but there is no interaction among these outputs. Further, we focus on a limit (small noise) where some analytic progress is possible. We will see that, even in this case, the optimal solutions have an interesting structure, which emerges as a result of the interplay between noise sources at the input and the output of the regulatory elements. For other approaches to the optimization of information transmission in biochemical and genetic networks, see Refs [43,44,45].
Optimization of information transmission is a concise
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