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Title: Modular Composition of Gene Transcription Networks}
Article:
Abstract:
Predicting the dynamic behavior of a large network from that of the composing modules is a central problem in systems and synthetic biology. Yet, this predictive ability is still largely missing because modules display context-dependent behavior. One cause of context-dependence is retroactivity, a phenomenon similar to loading that influences in non-trivial ways the dynamic performance of a module upon connection to other modules. Here, we establish an analysis framework for gene transcription networks that explicitly accounts for retroactivity. Specifically, a module's key properties are encoded by three retroactivity matrices: internal, scaling, and mixing retroactivity. All of them have a physical interpretation and can be computed from macroscopic parameters( dissociation constants and promoter concentrations) and from the modules' topology. The internal retroactivity quantifies the effect of intramodular connections on an isolated module's dynamics. The scaling and mixing retroactivity establish how intermodular connections change the dynamics of connected modules. Based on these matrices and on the dynamics of modules in isolation, we can accurately predict how loading will affect the behavior of an arbitrary interconnection of modules. We illustrate implications of internal, scaling, and mixing retroactivity on the performance of recurrent network motifs, including negative autoregulation, combinatorial regulation, two-gene clocks, the toggle switch, and the single-input motif. We further provide a quantitative metric that determines how robust the dynamic behavior of a module is to interconnection with other modules. This metric can be employed both to evaluate the extent of modularity of natural networks and to establish concrete design guidelines to minimize retroactivity between modules in synthetic systems.
Introduction:
We begin by introducing a standard mechanistic model for gene transcription networks, which includes protein production, decay, and reversible binding reactions between transcription factors( TFs) and promoter sites, required for transcriptional regulation. Specifically, transcription networks are usually viewed as the input/output interconnection of fundamental building blocks called transcriptional components. A transcriptional component takes a number of TFs as inputs, and produces a single TF as an output. The input TFs form complexes with promoter sites in the transcriptional component through reversible binding reactions to regulate the production of the output TF, through the process of gene expression( for details, see Methods). To simplify the notation, we treat gene expression as a one-step process, neglecting mRNA dynamics. This assumption is based on the fact that mRNA dynamics occur on a time scale much faster than protein production/decay[1]. In addition to this, including mRNA dynamics is not relevant for the study of retroactivity, and would yield only minor changes in our results( see Methods). Within a transcription network, we identify a transcriptional component with a node. Consequently, a transcription network is a set of interconnected nodes in which node represents the transcriptional component producing TF. There is a directed edge from node to if is a TF regulating the activity of the promoter controlling the expression of[29], in which case we call a parent of. Activation and repression are denoted by and, respectively. Modules are a set of connected nodes. Modules communicate with each other by having TFs produced in one module regulate the expression of TFs produced in a different module. When a node is inside the module, we call the corresponding TF an internal TF, while when node is outside the module we call the corresponding TF an external TF. Further, we identify external TFs that are parents to internal TFs as inputs to the module. Let, and denote the concentration vector of internal TFs, inputs and TF-promoter complexes, respectively. According to[30], we can write the dynamics of the module as( 1) where is the stoichiometry matrix and is the reaction flux vector. The reactions are either protein production/decay or binding/unbinding reactions. Therefore, we partition into and, representing the reaction flux vectors corresponding to production/decay and binding/unbinding reactions, respectively( see Methods). We assume that the DNA copy number is conserved, therefore, we can rewrite( 1) aswhere the upper left block matrix in is the zero matrix as DNA is not produced/degraded. As a result, with we obtain( 2) which we call the isolated dynamics of a module. Next, consider the case when the module is inserted into a network, which we call the context of the module. We represent all the quantities related to the context with an overbar. Let and denote the concentration vector of TFs and promoter complexes of the context, respectively. Furthermore, denote by and the reaction flux vectors corresponding to production/decay and binding/unbinding reactions between TFs and promoters in the context of the module, respectively. Then, the dynamics of the species in the module( and) and in the context( and) can be written as( 3) where the upper left block matrix is zero as DNA is assumed to be a conserved species. Furthermore, since and encapsulate the binding/unbinding reactions in the module and in its context, respectively, the off-diagonal block matrices in the upper right block matrix are zero. Similarly, as and encapsulate the production/decay reactions in the module and its context, respectively, the off-diagonal block matrices in the lower left block matrix are zero. Finally, the stoichiometry matrix represents how internal TFs of the module participate in binding/unbinding reactions in the context of the module( can be interpreted similarly). With describing the effective rate of change of due to intermodular binding reactions, we obtain( 4) which we call the connected dynamics of a module. We refer to as the retroactivity to the output of the module, encompassing retroactivity applied to the module due to the context of the module. Similarly, we call the retroactivity to the input of a module, representing retroactivity originating inside the module. The general interconnection of a group of modules can be treated similarly( Figure 1). As an example of the implications of retroactivity on the module's dynamic behavior, consider Figure 2. For the purpose of illustration, assume that and, external inputs to and( see Methods), are periodic( in general, they can be arbitrary time-varying signals). When the module is not connected to its context( Figure 2A), its output is periodic( Figure 2B). Upon interconnection with its context( Figure 2C), due to the retroactivity to the output applied by the context, the output of the module changes significantly( Figure 2D). Hence, connection with the context leads to a dramatic departure of the dynamics of the module from its behavior in isolation. This example illustrates that retroactivity significantly alters the dynamic behavior of modules after interconnection, therefore, it cannot be neglected if accurate prediction of temporal dynamics is required. Unfortunately, model( 4) provides little analytical insight into how measurable parameters and interconnection topology affect retroactivity. The aim of this paper is to provide a model that captures the effects of retroactivity, unlike standard regulatory network models of the same dimension based on Hill functions[1]. Specifically, we seek a model that explicitly describes the change in the dynamics of a module once it is arbitrarily connected to other modules in the network. This model is only a function of measurable biochemical parameters, TF concentrations, and interconnection topology.
Discussion:
In this paper, we have focused on retroactivity, one source of context-dependence, and demonstrated that the internal, scaling, and mixing retroactivity provide missing knowledge that captures loading effects due to intramodular and intermodular connections. The internal retroactivity quantifies the effect of intramodular load, applied by nodes within a module onto each other because of binding to promoter sites within the module. Given a module of interest, the effects of intramodular loading on the module's dynamics are captured by equations( 5) –( 6), in which one needs to replace the specific expressions of the Hill functions and node retroactivities provided in Figure 3, and the binary matrices encoding the network topology. The scaling retroactivity accounts for the intermodular loading that the context applies on a module, due to having some TFs of the module bound to promoter sites belonging to the context. The mixing retroactivity couples the dynamics of the module and that of the context upon interconnection, and it is non-zero when TFs from different modules bind non-independently at promoter sites. The effects of intermodular loading are captured by equations( 10) –( 13). To obtain this description, it is sufficient to consider the Hill functions and node retroactivities provided in Figure 3, together with the binary matrices, and representing the network topology. In general, the effects of the retroactivity matrices tend to increase with increased DNA copy number and/or decreased dissociation constants. We have illustrated that accounting for retroactivity reveals surprising dynamical properties of modules and, at the same time, can aid design. For example, negative autoregulation, depending on the gene copy number and production rates, can slow down the response of a system instead of speeding it up. A gene can respond to a perturbation applied to a different gene even in the absence of a regulatory path between the two genes. We have shown that this can occur when a group of TFs co-regulate common targets and these common targets are found in abundance. This type of motif, referred to as the dense overlapping regulon, is highly frequent in natural regulatory networks[1]. As a result, system identification techniques based on perturbation analysis[23] could erroneously identify non-existent regulatory linkages if retroactivity is not accounted for in the corresponding models. An activator-repressor clock on low copy DNA plasmid displays sustained oscillations when internal retroactivity is neglected, while oscillations are quenched once internal retroactivity is accounted for. However, by carefully adjusting the module's internal retroactivity through the addition of DNA load for the repressor, we can restore oscillations. A genetic toggle switch that can be flipped by a transient external stimulation requires a substantially longer stimulation to be flipped once it is connected to just few downstream targets. These facts are relevant, in particular, when designing synthetic biology circuits and multi-module systems. Similar to synthetic systems, natural systems are subject to retroactivity. For example, clocks responsible for circadian rhythms have a large number of downstream targets[44],[45], which, in turn, may apply substantial load. This load can affect the amplitude and frequency of oscillations of the clock as well as the stability of the corresponding limit cycle. This suggests that natural systems may have evolved to use retroactivity in advantageous ways such as using it to properly tune the dynamic behavior of a module without changing the module's components. This hypothesis is further supported by the fact that there are a large number of TF binding sites on the chromosome that do not have a regulatory function[46],[47]. These sites have an impact on the temporal response of TFs, and could therefore be exploited by nature to further control the dynamics of gene regulation. More generally, retroactivity provides means for information to travel from downstream targets to upstream regulators, therefore establishing indirect connections. In highly interconnected topologies, this information transfer can result in previously unknown ways of realizing sophisticated functions. One such example that we have provided is the adaptation function that topologies such as the dense overlapping regulon can realize by virtue of having nodes co-regulate multiple downstream targets. Based on the three retroactivity matrices, we provided a metric of robustness to interconnection, quantifying the percent change between the dynamics of a module in isolation and once connected to other modules. This metric is an explicit function of measurable parameters and becomes smaller when a module's internal retroactivity is large compared to the scaling retroactivity of the modules it connects to. This interplay may help uncover trade-offs in natural systems, providing a new angle for understanding natural principles of network organization. From an engineering perspective, we have provided quantitative design tools that can be employed in synthetic biology to appropriately match the internal and scaling retroactivity of connected circuits to preserve the circuits' behavior upon interconnection with different contexts. Our metric of robustness to interconnection further allows to evaluate the extent of modularity of a dynamical module, possibly enabling the discovery of previously unknown core processes. Our metric could be employed by currently available methods for partitioning networks into modules. Specifically, to evaluate connectivity, these methods rely on several metrics, for instance, edge betweenness[35], its extension to directed graphs with nonuniform weights[36], round trip distance[25] or retroactivity[37]. The metric of robustness to interconnection that we have introduced can enhance these methods by providing a way to evaluate modules on the basis of their functional robustness to interconnection in addition to distinguishing them at the connectivity level. The framework that we have proposed carries substantial conceptual analogies with the electrical circuit theory established by Thevenin[48], which has been used for more than a hundred years to analyze and to design electrical networks. Within this theory, each circuit has an equivalent input and output impedance( conceptually analogous to the scaling/mixing and internal retroactivity, respectively), and an equivalent energy source( playing a role similar to the isolated module dynamics). This theory has been instrumental for answering key questions in the analysis and design of electrical networks including, for example, how the output of a circuit changes after it is interconnected in a network; how to design circuits to maximize the power transfer upon connection( impedance matching); and how to design circuits whose input/output response is unaffected by loads. We believe that the framework proposed in this paper can be used in a similar way for the analysis and design of gene regulatory networks. Although our framework can be applied to a general gene transcription network, there are a number of aspects that it does not currently capture. These include post-translational protein modifications, such as phosphorylation, which are present in many regulatory networks and may potentially affect retroactivity. Including these will require to extend our framework to mixed gene transcription and signaling network models, leading to systems with multiple time scales. Furthermore, the transcription and translation processes use shared resources such as RNA polymerase and ribosomes, which may create couplings among unconnected circuits[17]. The dynamics of shared resources has not been included in our modeling framework and will be the focus of our future work.
Biological modules are inherently context-dependent as the input/output behavior of a module often changes upon connection with other modules . One source of context-dependence is retroactivity , a loading phenomenon by which a downstream system affects the behavior of an upstream system upon interconnection . This fact renders it difficult to predict how modules will behave once connected to each other . In this paper , we propose a general modeling framework for gene transcription networks to accurately predict how retroactivity affects the dynamic behavior of interconnected modules , based on salient physical properties of the same modules in isolation . We illustrate how our framework predicts surprising and counter-intuitive dynamic properties of naturally occurring network structures , which cannot be captured by existing models of the same dimension . We describe implications of our findings on the bottom-up approach to designing synthetic circuits , and on the top-down approach to identifying functional modules in natural networks , revealing trade-offs between robustness to interconnection and dynamic performance . Our framework carries substantial conceptual analogies with electrical network theory based on equivalent representations . We believe that the framework we have proposed , also based on equivalent network representations , can be similarly useful for the analysis and design of biological networks .