Stochastic Dynamics of Macromolecular-Assembly Networks
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Molecular Systems Biology (2006) doi:10.1038/msb4100061 & 2006 EMBO and Nature Publishing Group All rights reserved 1744-4292/06 www.molecularsystemsbiology.com 2006.0024 REVIEW Stochastic dynamics of macromolecular-assembly networks Leonor Saiz and Jose MG Vilar* 2003). These complexes form the backbone of the most funda- mental cellular processes, including gene regulation and Integrative Biological Modeling Laboratory, Computational Biology Program, signal transduction. Important examples are the assembly of Memorial Sloan-Kettering Cancer Center, New York, NY, USA the eukaryotic transcriptional machinery (Roeder, 1998), with * Corresponding author. Integrative Biological Modeling Laboratory, about hundred components, and the formation of signaling Computational Biology Program, Memorial Sloan-Kettering Cancer Center, complexes (Bray, 1998), with tens of different molecular 1275 York Avenue, Box #460, New York, NY 10021, USA. Tel.: þ 1 646 735 8085; Fax: þ 1 646 735 0021; E-mail: [email protected] species. One of the main challenges facing modern biology is to Received 28.11.05; accepted 3.3.06 move forward from the reductionist approach into the systemic properties of biological systems (Alon, 2003). A major goal is to understand how the dynamics of cellular processes emerges from the interactions among the different The formation and regulation of macromolecular com- molecular components. Typical computational approaches plexes provides the backbone of most cellular processes, approximate cellular processes by networks of chemical including gene regulation and signal transduction. The reactions between different molecular species (Endy and inherent complexity of assembling macromolecular struc- Brent, 2001). A strong barrier to this type of approaches is tures makes current computational methods strongly the inherent complexity of macromolecular complex forma- limited for understanding how the physical interactions tion. Complexes are typically made of smaller building blocks between cellular components give rise to systemic proper- with a modular organization that can be combined in a ties of cells. Here, we present a stochastic approach to study number of different ways (Pawson and Nash, 2003). The result the dynamics of networks formed by macromolecular of each combination is a specific molecular species and should complexes in terms of the molecular interactions of their be considered explicitly in a chemical reaction description. components. Exploiting key thermodynamic concepts, this Therefore, there are potentially as many reactions as the approach makes it possible to both estimate reaction rates number of possible ways of arranging the different elements, and incorporate the resulting assembly dynamics into the which grows exponentially with the number of the constituent stochastic kinetics of cellular networks. As prototype elements. Twenty components, for instance, already give rise systems, we consider the lac operon and phage k induction to over a million of possible species. switches, which rely on the formation of DNA loops by Two main types of avenues have been followed to tackle the proteins and on the integration of these protein–DNA exponential growth of the number of molecular species that complexes into intracellular networks. This cross-scale arise during macromolecular assembly. One is based on approach offers an effective starting point to move forward computer programs that generate reaction rate equations for from network diagrams, such as those of protein–protein all of the macromolecular species (Bray and Lay, 1997). The and DNA–protein interaction networks, to the actual other generates the different species dynamically (Morton- dynamics of cellular processes. Firth and Bray, 1998; Lok and Brent, 2005). Yet, none of the Molecular Systems Biology 16 May 2006; existing methods provides a consistent way to estimate the doi:10.1038/msb4100061 reaction rates. This obstacle is remarkable because the poten- Subject Categories: metabolic and regulatory networks; tial number of rates is even higher than the number of possible computational methods complexes. As a result, those methods often lead to unrealistic Keywords: computational methods; interaction networks; macro- situations, such as the formation of polymeric complexes that molecular complex assembly; regulatory cellular networks; do not exist under physiological conditions, which has been stochastic dynamics noted explicitly as intriguing caveats of the existing method- ologies (Bray and Lay, 1997; Lok and Brent, 2005). Here, we review the thermodynamic concepts under- lying macromolecular complex assembly and examine how they can be used to both derive the dynamics of complex Introduction formation and estimate the transition rates needed to build Cells consist of thousands of different molecular species that a faithful computational model. The resulting stochastic orchestrate their interactions to form extremely reliable approach does not give rise to the formation of unrealistic functional units (Hartwell et al, 1999). Such molecular complexes and addresses the exponential growth of the diversity and the pervasive ability of cellular components to number of species by stochastically exploring the set of establish multiple simultaneous interactions typically lead representative complexes. In this way, it brings the properties to the formation of large heterogeneous macromolecular of macromolecular interactions across scales up to the assemblies, also known as complexes (Pawson and Nash, dynamics of cellular networks. To illustrate the applicability & 2006 EMBO and Nature Publishing Group Molecular Systems Biology 2006 msb4100061-E1 Stochastic macromolecular assembly L Saiz and JMG Vilar of this approach, we focus on DNA–protein complexes and of multistate proteins to receptor docking sites (Borisov their integration in gene regulatory networks. We consider as et al, 2005), and signaling through clusters of receptors prototype systems the induction switches in the lac operon (Bray et al, 1998; Duke and Bray, 1999). In practice, current (Mu¨ller-Hill, 1996) and phage l (Ptashne, 2004), the two binary-variable approaches have strong limitations to tackle systems that led to the discovery of gene regulation (Jacob and the assembly of macromolecular complexes. On the one Monod, 1961). hand, there are combinations of variables that do not have a physical existence. Explicitly, if a component that bridges two disconnected parts of the complex is missing, then the Representation of macromolecular complex does not exist and if two components occupy the complexes same position, they cannot be present within the complex simultaneously. On the other hand, the structure of the A crucial aspect is to use a description for macromolecular complex does not have to remain fixed. A complex can have complexes that can capture the underlying complexity different conformations and the components can be present in in simple terms. It is possible to take advantage of the fact several states with different properties. None of the existing that macromolecular complexes have typically unambiguous approaches based on binary variables incorporates all these structures, where only certain molecular species can occupy key features needed to study macromolecular complex a given position within the complex. In such cases, the specific formation. In the following section, we exploit the underlying configuration or state of the macromolecular complex can be thermodynamics to put forward a binary description appli- y y described by a set of M variables, denoted by s¼(s1, si, sM), cable to macromolecular assembly. whose values indicate whether a particular molecular compo- nent is present or not at a specific position. We chose si¼1to Thermodynamics of assembly indicate that the component is present and si¼0 to indicate that it is not present (Figure 1A). With this description, the Thermodynamics allows for a straightforward connection potential number of specific complexes is 2M and the number of the binary description with the molecular properties of 1 M M E 2MÀ1 of reactions 22 (2 À1) 2 . the system. Each configuration of the macromolecular com- The use of binary variables provides a concise method to plex has a corresponding free energy, which is a quantity describe all the potential complexes without explicitly enumer- that indicates the tendency of the system to change its state. ating them. This type of approach has been used in a wide range Transformations that decrease the free energy of the system of interesting biological situations, such as diverse allosteric are favored over those that increase it; that is, the tendency processes (Bray and Duke, 2004), binding of molecules to of the system is to evolve towards the lowest free energy a substrate (Keating and Di Cera, 1993; Di Cera, 1995), binding minimum. The statistical interpretation of thermodynamics (Gibbs, 1902; Hill, 1960) connects the equilibrium probability Ps of s DG s AB state with its free energy ( ) through the expression A 15 1 B ÀDGðsÞ=RT 15 Ps ¼ e ; C Z P ÀDG(s)/RT o sA=1, sB=0, sC=1 ∆G = 30 where Z¼ s e is the normalization factor and RT is the gas constant times the absolute temperature (RTE 0.6 kcal/mol for typical experimental conditions). CD–20 15 15 One crucial aspect is that the free energy of any given y y –20 –20 15 configuration, s¼(s1, ,si ,sM), can be obtained to any 15 15 15 degree of accuracy by expanding the free energy in powers of –20 –20 the binary variables: ∆G o= 5 ∆G o= –15 XM