Negative feedback and physical limits of genes

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Abstract

This paper compares the auto-repressed gene to a simple one (a gene without auto-regulation) in terms of response time and output noise under the assumption of fixed metabolic cost. The analysis shows that, in the case of non-vanishing leak expression rate, the negative feedback reduces both the switching on and switching off times of a gene. The noise of the auto-repressed gene will be lower than the one of the simple gene only for low leak expression rates. Summing up, for low, but non-vanishing leak expression rates, the auto-repressed gene is both faster and less noisier compared to the simple one.

Highlights

► We compare an auto-repressed gene to a simple gene under equal metabolic cost. ► For non-zero leak rate, the auto-repressed gene is faster in switching on and off. ► Increasing leaky expression increases the output noise. ► For high leak rate the auto-repressed gene is noisier but faster. ► For zero leak rate the auto-repressed gene is slower but less noisy.

Introduction

Genes which maintain a functional relationship between the concentration of the regulatory input protein(s) and the concentration of the output protein can be thought of as being capable of performing computations (Weiss et al., 2003, Buchler et al., 2003, Yokobayashi et al., 2002, Mayo et al., 2006, Fernando et al., 2009). For instance, often cells need to respond to the presence or the absence of various chemical factors. In this case, one can say that the cell performs some sort of “computations” on the input chemical factors and, based on the result of these computations, the cell will produce a specific chemical or physical response (output). A classic example is the lac operon in Escherichia coli where the proteins associated with the lactose metabolism are produced only when the glucose is absent and the lactose is present (Setty et al., 2003). This is often approximated by an AND NOT gate, which suggests that the operon performs logical computations.

These “computations” performed by genes are characterised by several properties (such as accuracy, speed and energy cost) which are significantly influenced by the specificity of the environment (the cell). For instance, in the case of low number of molecules, inherent fluctuations in reaction rates, caused by thermal noise, induce stochastic fluctuations (noise) in the copy number of molecules (Lei, 2009). Usually, in living cells, there are few copies of mRNA molecules, and one or two copies per gene (Arkin et al., 1998). Consequently, the gene expression process is affected by noise (Kaern et al., 2005). In the context of genes as computational units, stochastic fluctuations can hide useful signals in noise and, thus, the accuracy of the response is reduced.

In addition to accuracy, computations are also characterised by the speed at which they are performed (Bennett, 1982). The response of a gene to a change of input is not instantaneous, but rather affected by a time delay. This time delay is often called the response time of the gene (Alon, 2007a) and is connected to the speed at which the genes “compute”, in the sense that higher response times translate in slower computations, while lower response times in faster computations. In the case of genes, the speed at which a gene responds to changes in the transcription factors abundance is of high importance. In particular, a cell that is able to respond faster to changes in the environment can have certain advantages over slower cells. For example, a cell that is able to uptake food faster can consume more nutrients compared to a slower cell and, consequently, have an energy advantage over it.

Ideally, one would want to increase both speed and accuracy as much as possible, but this is often limited by the available energy supply (Bennett, 1973, Bennett, 1982, Lloyd, 2000). Each cellular process (protein production, protein decay and maintenance processes) has a metabolic cost attached to it, which is, usually, measured in number of ATP molecules (Akashi and Gojobori, 2002). The notion of cost used in this paper is not the exact quantitative measure of the actual metabolic cost, but rather a number which describes how the actual metabolic cost scales when the parameters of the genes are changed.

With few exceptions, these three properties (speed, accuracy and cost) were investigated previously in a stand-alone fashion. These exceptions include studies which analysed only the speed and accuracy and disregarded the cost in several molecular systems, such as DNA based logic gates (Stojanovic and Stefanovic, 2003), protein–protein interaction networks (Wang et al., 2010) and gene regulatory networks (Rosenfeld et al., 2005, Isaacs et al., 2005, Hooshangi et al., 2005, Shahrezaei et al., 2008). A few other studies examined all three properties (speed, accuracy and cost) in various gene regulatory networks, like auto-repressed genes (Stekel and Jenkins, 2008), toggle switches (Mehta et al., 2008) or gene networks that used frequency encoded signals (Tan et al., 2007). Nevertheless, these studies which integrate all three properties (speed, accuracy and cost) addressed different scenarios and used a different measure of cost compared to the one used in this contribution (for a discussion on the measure of cost see below). For a comprehensive review on computational properties of molecular systems see Zabet (2010).

In addition to the aforementioned studies, Zabet and Chu (2010) showed that, in the case of a simple gene (a gene without auto-regulation), the speed, accuracy and cost properties are interconnected. In particular, they found that under fixed metabolic cost there is a speed–accuracy trade-off, which is controlled by the decay rate, i.e., high decay rates lead to faster and less accurate responses while lower decay rates to slower and more accurate ones.

One of the central results of this previous work by Zabet and Chu (2010) stated that the speed–accuracy trade-off is optimal for systems with zero leak rates, i.e., there are no solutions that have better speed–accuracy characteristics. The vanishing leak expression rate represents a theoretical performance limit of a gene under fixed cost, but it is difficult to achieve in real systems and would require high metabolic cost (Zabet and Chu, 2010). The current paper will investigate whether the performance of a gene can be improved beyond this optimal configuration (of zero leak rate), i.e., if a gene can display faster response times and less noise at the output without increasing the metabolic cost.

One candidate mechanism to enhance the performance of a gene is the negative feedback, which is a network motif in bacterial cells (Savageau, 1974, Thieffry et al., 1998, Shen-Orr et al., 2002, Alon, 2007a), in the sense that it is a sub-network which is encountered with high occurrence, e.g. in E. coli 40% of the genes are auto-repressed (Austin et al., 2006). The fact that it is encountered with high occurrence suggests that auto-repressed genes have certain advantages compared to simple ones. This paper aims to investigate whether negative feedback can enhance both response time and output noise while keeping the metabolic cost fixed (equal to the one of the gene without auto-repression).

The auto-repressed gene is a well studied system, which received great attention from the community over the past decade (Alon, 2007b). Rosenfeld et al. (2002) showed that negative feedback reduces only the response time of switching on (when the gene goes from a low expression to a high one). In the current setting (genes as computational units), the switching direction is not important and, thus, the mechanism should be capable of reducing the response times of both switching on and switching off (when the gene goes from a high expression to a low one). One of the assumptions of the aforementioned study of Rosenfeld et al. (2002) is that genes do not display leaky expression, which is obviously not true for all genes. Actually, most of the genes will display non-zero leak expression rates because the metabolic cost associated with removing the leak rate would be very high (Zabet and Chu, 2010). Thus, this paper aims to investigate whether, under the assumption of leaky expression, the negative feedback can reduce the response time of both switching on and off.

Furthermore, experimental evidence suggested that a negatively auto-regulated gene displays lower noise compared with the gene without any type of auto-regulation (Becskei and Serrano, 2000). Analytical results confirmed that the noise of the auto-repressed gene is lower than the one of the simple gene (Thattai and van Oudenaarden, 2001, Paulsson, 2004). However, these two studies (Thattai and van Oudenaarden, 2001, Paulsson, 2004) did not consider fixed metabolic cost. Other studies derived the equation of noise analytically under the assumption that the simple gene and the auto-repressed one display equal average number of molecules at steady state (Paulsson and Ehrenberg, 2000, Stekel and Jenkins, 2008, Zhang et al., 2009). Their results confirmed that the auto-repressed gene reduces the output noise. Nevertheless, they assumed that if the two systems have an equal average number of molecules at steady state, they also have an equal metabolic cost.

A better measure for the metabolic cost is the production rate of a gene (Zabet and Chu, 2010, Chu et al., 2011). This is justified by the fact that a measure of metabolic cost should describe the energy consumption per time unit. For example, consider the case of two proteins (X1 and X2) that have the same average number of molecules at steady state, but the first one (X1) is produced and decayed faster compared with the second one (X2). Then, more molecules of the first protein (X1) will be produced and decayed compared with the second one (X2) and, consequently, the metabolic cost associated with the first protein (X1) will be higher compared with the one of the second protein (X2). Hence, the production rate will describe better the scaling properties of the metabolic cost compared to the average number of molecules at steady state.

This paper aims to compare systems that have equal metabolic cost. In the case of fixed decay rate, imposing the production rates (the measure of metabolic cost) of two systems to be equal leads to the output steady state abundances of the systems to be equal as well. This means that, when production rate is kept fixed, it is ensured that a previously used measure of cost (steady state abundance) is also kept fixed. Nevertheless, the current analysis is not limited to cases where the decay rate is kept fixed (see for example Fig. 6) and, in those cases, attempting to keep production rates fixed will lead to variable steady state abundances. Due to the reasons mentioned above, the production rate will be used as the only indicator of metabolic cost when the decay rate is not fixed. Note however that, for fixed decay rate, the two measures are equivalent and, consequently, the results obtained under the assumption of fixed production rates (metabolic cost) are also valid for equal steady state abundances.

The results from this contribution show that negative feedback reduces the response time in the case of leaky expression for both switching on and switching off. In addition, for low leak rates, negative feedback reduces the noise, while for high leak rates it increases the noise. Both these results were obtained under the assumption of fixed metabolic cost. Furthermore, the analysis identified a subset in the parameter space (low but non-vanishing leak rates) where the negatively auto-regulated gene outperforms the simple one in both speed and accuracy, thus, setting a new theoretical performance limit for genes.

Section snippets

Simple gene

The model of the simple gene consists of a single gene Gy, which has an output y and is regulated by a single input species x; see Fig. 1(a). This system is described by the following set of chemical reactions:βf(x)y,yμHere, β is the maximal expression rate of the gene, f(x) is the regulation function, x is the concentration of the regulatory input, and μ is the degradation rate of the product of the gene; see Table 1.

Gene regulation functions are often approximated by Hill functions (

Response time

Generally, one would want to process information as fast as possible, but genes are very slow, in the sense that the time required to turn on/off a gene (the switching time) is of the order of tens of minutes, even for an instant input change. Thus, it is essential to investigate what constrains the speed at which genes function and whether there are any methods to increase this speed.

A common measure of the processing speed of genes is the response time that is the time required for the output

Noise

Gene expression is affected by noise (Spudich and Koshland, 1976, Arkin et al., 1998, Elowitz et al., 2002). This noise is a consequence of the fact that genes have low copy numbers and that they are slowly expressed (Kaern et al., 2005). In the context of genes as computational units, this output noise is undesirable because it makes difficult the assessment of the output of the gene as either low or high.

At steady state one can compute the variance of the output species y of the simple gene

Speed–accuracy trade-offs

Recently, Zabet and Chu (2010) showed that the processing speed and accuracy of genes are connected, in the sense that there is a trade-off between speed and accuracy, which is controlled by the decay rate. Fig. 6 confirms the existence of this trade-off also in the case of the auto-repressed gene. Furthermore, the graph confirms that for low leak rates the auto-repressed gene displays a better trade-off curve compared to the simple one; see Fig. 6(a). Note that not only the metabolic cost is

Discussion

Negative auto-regulation was suggested as an alternative approach to reduce the response time of a single gene. Rosenfeld et al. (2002) showed theoretically that negative auto-regulation can speed up only the turn on response time, i.e., the turn on time of a negatively auto-regulated gene is five times smaller than the one of a simple gene. In the context of genes as computational units, the direction of switching is not important in the sense that the system needs to turn both on and off as

Acknowledgements

The author would like to thank Dr. Dominique F. Chu, Professor Andrew N.W. Hone, Dr. Boris Adryan and the anonymous reviewers for useful comments which lead to improvements of the manuscript and Felicia Dana Zabet for proofreading the manuscript.

References (53)

  • A. Arkin et al.

    Stochastic kinetic analysis of developmental pathway bifurcation in phage l-infected Escherichia coli cells

    Genetics

    (1998)
  • D.W. Austin et al.

    Gene network shaping of inherent noise spectra

    Nature

    (2006)
  • A. Bar-Even et al.

    Noise in protein expression scales with natural protein abundance

    Nat. Genet.

    (2006)
  • A. Becskei et al.

    Engineering stability in gene networks by autoregulation

    Nature

    (2000)
  • C.H. Bennett

    Logical reversibility of computation

    IBM J. Res. Dev.

    (1973)
  • C.H. Bennett

    The thermodynamics of computation—a review

    Int. J. Theor. Phys.

    (1982)
  • F.J. Bruggeman et al.

    Noise management by molecular networks

    PLoS Comput. Biol.

    (2009)
  • N.E. Buchler et al.

    On schemes of combinatorial transcription logic

    Proc. Nat. Acad. Sci.

    (2003)
  • M.B. Elowitz et al.

    Stochastic gene expression in a single cell

    Science

    (2002)
  • C.T. Fernando et al.

    Molecular circuits for associative learning in single-celled organisms

    J. R. Soc. Interface

    (2009)
  • F. Hayot et al.

    The linear noise approximation for molecular fluctuations within cells

    Phys. Biol.

    (2004)
  • S. Hooshangi et al.

    Ultrasensitivity and noise propagation in a synthetic transcriptional cascade

    Proc. Nat. Acad. Sci.

    (2005)
  • G. Hornung et al.

    Noise propagation and signaling sensitivity in biological networks: a role for positive feedback

    PLoS Comput. Biol.

    (2008)
  • F.J. Isaacs et al.

    Signal processing in single cells

    Science

    (2005)
  • M. Kaern et al.

    Stochasticity in gene expression: from theories to phenotypes

    Nat. Rev. Genet.

    (2005)
  • S. Lloyd

    Ultimate physical limits to computation

    Nature

    (2000)
  • Cited by (0)

    1

    Present address: Cambridge Systems Biology Centre and Department of Genetics, University of Cambridge, Tennis Court Road, Cambridge CB2 1QR, UK.

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