Robustness, Stability and Efficiency of Phage Λ Genetic Switch: Dynamical Structure Analysis

Computational studies of biological networks can help to identify components and wirings responsible for observed phenotypes. However, studying stochastic networks controlling many biological processes is challenging. Similar to Schrödinger's equation in quantum mechanics, the chemical master equation (CME) provides a basic framework for understanding stochastic networks. However, except for simple problems, the CME cannot be solved analytically. Here we use a method called discrete chemical master equation (dCME) to compute directly the full steady-state probability landscape of the lysogeny maintenance network in phage lambda from its CME. Results show that wild-type phage lambda can maintain a constant level of repressor over a wide range of repressor degradation rate and is stable against UV irradiation, ensuring heritability of the lysogenic state. Furthermore, it can switch efficiently to the lytic state once repressor degradation increases past a high threshold by a small amount. We find that beyond bistability and nonlinear dimerization, cooperativity between repressors bound to O R 1 and O R 2 is required for stable and heritable epigenetic state of lysogeny that can switch efficiently. Mutants of phage lambda lack stability and do not possess a high threshold. Instead, they are leaky and respond to gradual changes in degradation rate. Our computation faithfully reproduces the hair triggers for UVinduced lysis observed in mutants and the limitation in robustness against mutations. The landscape approach computed from dCME is general and can be applied to study broad issues in systems biology.B acteriophage lambda is a virus that infects Escherichia coli cells. It has served as a model system for studying regulatory networks and for engineering gene circuits (1-5). Of central importance is the molecular circuitry that controls phage lambda to choose between two productive modes of development, namely, the lysogenic state and the lytic state (Fig 1A). In the lysogenic state, phage lambda represses its developmental function, integrates its DNA into the chromosome of the host E. coli bacterium, and is replicated in cell cycles for potentially many generations. When threatening DNA damage occurs, phage lambda switches from the epigenetic state of lysogeny to the lytic state and undergoes massive replications in a single cell cycle, releases 50-100 progeny phages upon lysis of the E. coli cell. This switching process is called prophage induction (5).The molecular network that controls the choice between these two different physiological states has been studied extensively during the past 40 y (5-9). All of the major molecular components of the network have been identified, binding constants and reaction rates characterized, and there is a good experimental understanding of the general mechanism of the molecular switch (5). Theoretical studies have also contributed to the illumination of the central role of stochasticity (3) and the stability of lysogen against spontaneous switching (4, 10). With the advent o...

Section: Probability Landscape Of Phage Lambda In Lysogenic and Lyticsupporting

confidence: 73%

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confidence: 99%

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Probability landscape of heritable and robust epigenetic state of lysogeny in phage lambda

Cao

Liang

2010

“…For example, in their analysis of the robustness of phage 's lysogenic phase (55), Aurell et al (56) and Zhu et al (57) assumed that monomers and dimers were degraded with equal rates and concluded that additional source(s) of cooperativity are needed to explain the observed robustness. We suggest that cooperative stability might be another possible source of cooperativity that needs to be examined critically.…”

Section: Discussionmentioning

confidence: 99%

Nonlinear protein degradation and the function of genetic circuits

Buchler

Gerland

Hwa

2005

159

170

The functions of most genetic circuits require a sufficient degree of cooperativity in the circuit components. Although mechanisms of cooperativity have been studied most extensively in the context of transcriptional initiation control, cooperativity from other processes involved in the operation of the circuits can also play important roles. In this work, we examine a simple kinetic source of cooperativity stemming from the nonlinear degradation of multimeric proteins. Ample experimental evidence suggests that protein subunits can degrade less rapidly when associated in multimeric complexes, an effect we refer to as ''cooperative stability.'' For dimeric transcription factors, this effect leads to a concentration-dependence in the degradation rate because monomers, which are predominant at low concentrations, will be more rapidly degraded. Thus, cooperative stability can effectively widen the accessible range of protein levels in vivo. Through theoretical analysis of two exemplary genetic circuits in bacteria, we show that such an increased range is important for the robust operation of genetic circuits as well as their evolvability. Our calculations demonstrate that a few-fold difference between the degradation rate of monomers and dimers can already enhance the function of these circuits substantially. We discuss molecular mechanisms of cooperative stability and their occurrence in natural or engineered systems. Our results suggest that cooperative stability needs to be considered explicitly and characterized quantitatively in any systematic experimental or theoretical study of gene circuits.amplification ͉ dimerization ͉ bistability ͉ oscillation I t is widely recognized that controlled proteolysis, where the degradation of one protein depends on the presence of another protein in the cell, can play an important regulatory role in genetic circuits (1). Here, we examine another effect of proteolysis that does not involve such regulatory control, but can nevertheless impact the function of genetic circuits in important ways. It is a kinetic, cooperative effect predicated on the following two essential ingredients: (i) the fact that many proteins perform their physiological functions as dimers or higher-order oligomers, and (ii) the tendency for the oligomers to be more stable (to proteolysis) than their monomeric components. This effect, referred to below as ''cooperative stability,'' has been discussed previously in qualitative terms in the context of many well-studied examples in prokaryotes and eukaryotes (1, 2). For example, in the SOS response of Escherichia coli, UmuC degradation is rescued by oligomerization with UmuDЈ 2 (3). Additionally, in Saccharomyces cerevisiae, the dimerization of a1 and ␣2 reduced the degradation rate by as much as 15-fold (4). Possible molecular mechanisms that give rise to cooperative stability include enhanced thermal stability of proteins upon mutual association [because thermal instability correlates with the rate of degradation (5, 6)] and the burial of proteolytic recogni...

“…Experimental and theoretical studies are steadily advancing our knowledge of the wiring and input-output characteristics of a variety of natural and designed biological networks. These efforts have focused on characterizing the components and interactions for specific biological networks and their dynamical behaviors, by using standard genetic and biochemical approaches in conjunction with mathematical analysis of discovered circuits (1)(2)(3)(4)(5)(6)(7). Valuable insights into certain design features of biological networks have emerged through these efforts (8)(9)(10)(11)(12)(13) and are used to guide the design of synthetic systems (14)(15)(16).…”

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Highly designable phenotypes and mutational buffers emerge from a systematic mapping between network topology and dynamic output

Nochomovitz

2006

Deciphering the design principles for regulatory networks is fundamental to an understanding of biological systems. We have explored the mapping from the space of network topologies to the space of dynamical phenotypes for small networks. Using exhaustive enumeration of a simple model of three-and four-node networks, we demonstrate that certain dynamical phenotypes can be generated by an atypically broad spectrum of network topologies. Such dynamical outputs are highly designable, much like certain protein structures can be designed by an unusually broad spectrum of sequences. The network topologies that encode a highly designable dynamical phenotype possess two classes of connections: a fully conserved core of dedicated connections that encodes the stable dynamical phenotype and a partially conserved set of variable connections that controls the transient dynamical flow. By comparing the topologies and dynamics of the three-and four-node network ensembles, we observe a large number of instances of the phenomenon of ''mutational buffering,'' whereby addition of a fourth node suppresses phenotypic variation amongst a set of three-node networks.designability ͉ dynamical phenotype ͉ enumeration ͉ mutational buffering ͉ regulatory network D iscerning the structure and function of cellular networks is essential to the development of a true understanding of biological systems. Experimental and theoretical studies are steadily advancing our knowledge of the wiring and input-output characteristics of a variety of natural and designed biological networks. These efforts have focused on characterizing the components and interactions for specific biological networks and their dynamical behaviors, by using standard genetic and biochemical approaches in conjunction with mathematical analysis of discovered circuits (1-7). Valuable insights into certain design features of biological networks have emerged through these efforts (8-13) and are used to guide the design of synthetic systems (14-16). The choice of which synthetic circuits to build is often a matter of careful hand-picking guided by experimental restrictions.Reverse-engineering and modeling of specific experimental systems on a case-by-case basis is necessary and meritorious. However, the space of networks and associated dynamics is potentially very large, and parallel approaches that consider broad ensembles of networks may advance our understanding of general design principles in ways that the serial strategies may have difficulty revealing. Therefore, we have chosen to explore general design principles by using a global strategy. We expect that exploration of an entire ensemble of networks and associated dynamics will reveal statistical signatures connecting network architectures to categories of dynamical phenotypes. In particular, we analyze the relationship between the space of topology and the space of dynamics by employing an analogy to the protein ''designability principle,'' which states that compact protein structures that can be encoded by a wide array of ...