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.
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.
Here, we focus on a single module without inputs and describe how retroactivity among nodes, modeled by in (2), affects the module' s dynamics. To this end, we provide a model that well approximates the isolated module dynamics, in which only measurable macroscopic parameters appear, such as dissociation constants and TF concentrations. We then present implications of this model for negative autoregulation, combinatorial regulation and the activator-repressor clock of [18]. Employing assumptions (a) – (b), we obtain the first main result of the paper as follows. Let denote the vector of concentrations of internal TFs, then the dynamics (5) well approximate the dynamics of in (2) in the isolated module with (6) where represents external perturbations to (inducer, noise, or disturbance, unless specified otherwise), is the decay rate of, and is the Hill function modeling the production rate of, regulated by the parents of. We call the retroactivity of node. For the most common binding types, Figure 3 shows the expressions of and (for their definition, see Methods). The binary matrix has as many columns as the number of nodes in the module, and as many rows as the number of parents of, such that its element is 1 if the parent of is, otherwise the entry is zero. That is, an entry in the following matrixis 1 if the species indexing the corresponding row and column are the same, otherwise the entry is zero, yielding. Finally, is the set of nodes having parents from inside the module. For the derivation of this result, see Theorem 1 in Supporting Text S2. We call the internal retroactivity of the module as it describes how retroactivity among the nodes internal to the module affects the isolated module dynamics. When, we have, the commonly used Hill function-based model for gene transcription networks [3]. It is possible to show that represents the rate of change of total (free and bound) TFs (see Supporting Text S2). Hence, (6) describes how changes in the total concentration of TFs relate to changes in the concentration of free TFs. Specifically, to change the concentration of free TFs by one unit, the module has to change the total concentration of TFs by units, as units are “spent on” changing the concentration of bound TFs. Having implies that the module' s effort on affecting the total concentration of TFs is entirely spent on changing the concentration of free TFs. By contrast, implies that no matter how much the total concentration of TFs changes, it is not possible to achieve any changes in the free concentration of some of the TFs. Therefore, the internal retroactivity describes how “stiff” the module is against changes in due to loading applied by internal connections. The entries of have the following physical interpretation. Consider first a module with the autoregulated node, that is, has a single parent: itself. The retroactivity of node is, where is given in Figure 3. In this case, we obtain by (6), so that (5) yields. Hence, the greater, the harder to change the concentration of free by changing its total concentration (the “stiffer” the node), and the temporal dynamics of become slower. The retroactivity of node can be increased by increasing its DNA copy number or by decreasing the dissociation constant of. For a node with two parents, we provide the explicit formula for in Figure 3 in the case of the most frequent binding types, so that here we simply write (7) The diagonal entries and in (7) can be interpreted similarly to, while the off-diagonal entries can be interpreted as follows. Having means that the second parent facilitates the binding of the first, whereas represents blockage (can be interpreted similarly with the parents having reverse roles). Therefore, we have in the case of independent binding (Figure 3), as the parents bind to different sites. By contrast, we have in the case of competitive binding (Figure 3), since the parents are competing for the same binding sites, forcing each other to unbind. Following a similar reasoning, we obtain in the case of cooperative binding (Figure 3). Notice that is scaled by the total concentration of promoter, which can be changed, for example, in synthetic circuits by changing the plasmid copy number. In order to illustrate the effects of intramodular connections, we consider three recurrent network motifs in gene transcription networks: (i) negative autoregulation of a gene, (ii) combinatorial regulation of a gene by two TFs, and (iii) the activator-repressor clock of [18]. After investigating how retroactivity due to intramodular connections affect a single module' s dynamics, we next determine how the dynamics of a module change when the module is inserted into its context. To this end, we first extend the model in (5) to the case in which the module has external TFs as inputs. Hence, let denote the concentration vector of TFs external to the module. With this, we obtain that the dynamics (10) well approximate the dynamics of in (2) with (11) where is the set of nodes having parents from outside the module (external TFs), and the binary matrix has as many columns as the number of inputs of the module, and as many rows as the number of parents of, such that its element is 1 if the parent of is, otherwise the entry is zero. That is, an entry in the following matrixis 1 if the species indexing the corresponding row and column are the same, otherwise the entry is zero, yielding. Furthermore, note that in the presence of input, both and given in (6) depend on and, as some of the parents of internal TFs are external TFs. For the derivation of this result, see Theorem 2 in Supporting Text S2. Before stating the main result of this section, we first provide the interpretation of. Recall that implies that the total concentrations of internal TFs are constant. In this case, (10) reduces to, where is the concentration vector of free internal TFs. This means that the concentrations of free internal TFs can still be changed subsequent to changes in the external TFs (input), despite the fact that the total concentration (free and bound) of internal TFs remains unaffected. Therefore, captures the phenomenon by which external TFs force internal TFs to bind/unbind, for instance, by competing for the same binding sites. Having means that external TFs do not affect the binding/unbinding of internal TFs, which is the case, for example, when all bindings are independent. Thus, we call the external retroactivity of the module. The main result of this section describes how the context of a module affects the module' s dynamics due to retroactivity. Specifically, we consider the module of interest and we represent the rest of the network, the module' s context, as a different module. As previously, we use the overbar to denote that a quantity belongs to the context. With this, we obtain that the dynamics (12) well approximate the dynamics of and in (4) in the module connected to the context with (13) where and denote the number of nodes in the module and in the context, respectively. Furthermore, the binary matrix has as many rows as the number of inputs of the module, and as many columns as the number of nodes in the context, such that its element is 1 if the input of the module is the internal TF of the context (), otherwise the entry is zero. That is, an entry in the following matrixis 1 if the species indexing the corresponding row and column are the same, otherwise the entry is zero, yielding. The quantities corresponding to the context, that is, , and are defined similarly with the only difference that variables with and without overbar have to be swapped (for instance, and have to be swapped in (13) ). For the derivation of this result, see Theorem 3 in Supporting Text S2. We next provide the interpretation of the scaling and mixing retroactivity. The reduced order model (12) describes how retroactivity between the module and the context affects each other' s dynamics. Note that zero matrices, , and lead to no alteration in the dynamics upon interconnection. To further deepen the implications of these matrices and their physical meaning, note that when, the dynamics of the module after interconnection become (14) that is, determines how the isolated dynamics of the module get “scaled” upon interconnection. Therefore, we call the scaling retroactivity of the context, accounting for the loading that the context applies on the module as some of the TFs of the module are taken up by promoter complexes in its context (we obtain if nodes in the context do not have parents in the module, that is, if). Since the dynamics of the context enter into the module' s dynamics through, we call the mixing retroactivity of the context, referring to the “mixing” of the dynamics of the module and that of its context. The mixing retroactivity establishes how internal TFs force external TFs to bind/unbind, so that can be ensured if the binding of parents from the module is independent from that of the parents from the context. This holds if nodes in the context are not allowed to have parents in both the module and in the context (). When, a perturbation applied in the context can result in a response in the upstream module, even without TFs in the context regulating TFs in the module, leading to a counter-intuitive transmission of signals from downstream (context) to upstream (module). With this, we can explain the simulation results in Figure 2D by analyzing and for the system in Figure 2C. Let and let be defined as in (7), where, , , and are given in Figure 3. Then, we have by (13) and and by (6). Hence, expression (12) yields (15) describing the dynamics of upon interconnection with its context, where and describe the dynamics of and, respectively, when the module and the context are not connected to each other. If, then (15) reduces tothat is, the context rescales the dynamics of the module. The smaller, that is, the greater the scaling retroactivity, the greater the effect of this scaling. Note that since the scaling factor is smaller than 1 (unless the scaling retroactivity is zero, i. e. ,), the effect of the scaling retroactivity of the context in this case is to make the temporal dynamics of the module slower. Once, in addition to this sclaing effect, the dynamics of the context appear in the dynamics of the module (Figure 2D). Referring to (15), we can quantify the effect of the context on the module, considering the ratio. The greater, that is, the greater, the stronger the contribution of the context compared to that of the module to the dynamics of the module upon interconnection. Here, both and increase, for instance, with the copy number of (Figure 3). Connecting the module to its context such that and are competing for the same binding sites is less desirable than employing independent binding, as the dynamics of the context (downstream system) can suppress the dynamics of the module (upstream system). Dismantling the dynamics of the module will “misinform” other downstream systems in the network that are regulated by. From a design perspective, multi-module systems should be designed and analyzed such that the modules have zero mixing retroactivity. This can be achieved, for instance, by avoiding non-independent binding at the interface nodes (at in Figure 2B). However, since completely independent binding can be hard to realize in the case of combinatorial regulation, nodes integrating signals from different modules should not be placed into the input layer (nodes having parents from other modules), yielding. This can be achieved by introducing an extra input node in the downstream module (see Supporting Figure S1). Next, we quantify the difference between the isolated and connected module behavior. In particular, we provide a metric of the change in the dynamics of a module upon interconnection with its context, dependent on and and under the assumption that (obtained, for instance, by avoiding mixing parents from the module and the context). The isolated dynamics of the module can be well approximated by the reduced order model in (10), and let denote its solution. Once we connect the module to its context, its dynamics change according to (14), which we write as and let denote the corresponding solution. Using the sub-multiplicative property of the induced 2-norm, we have that the percentage change of the dynamics upon interconnection can be bounded from above as follows: (16) Furthermore, with independent of and, such that, we obtain thatthat is, provides an upper bound on the percentage change in the dynamics of the module, and on the difference in the trajectories of the module upon interconnection with its context. Furthermore, by using the properties of the induced 2-norm, we obtain that we can pickprovided that for all and, where and denote the smallest singular value of and largest singular value of, respectively. For the mathematical derivations, see Supporting Text S2. This suggests that the module becomes more robust to interconnection by increasing or by decreasing. Such a metric can be used both in the analysis and in the design of complex gene transcription networks as follows. Given any network and a desired module size (number of nodes within the module), we can identify the module that has the least value of, that is, the module with the greatest guaranteed robustness to interconnection. Furthermore, we can also evaluate existing partitionings based on other measures (e. g. , edge betweenness [35], its extension to directed graphs with nonuniform weights [36], round trip distance [25] or retroactivity [37]) with respect to robustness to interconnection. From a design point of view, one can design multi-module systems such that internal, scaling and mixing retroactivities allow for low values of, leading to modules that behave almost the same when connected or isolated. We next illustrate the effect of intermodular connections on the dynamics of interconnected modules, considering both a synthetic genetic module that is being employed in a number of applications and a natural recurring network motif.
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.
The production of TF is regulated by its parents: they bind to the promoter of, and form complexes, with the promoter. Each of these complexes, in turn, produce with a different rate, where we use a one-step production process encapsulating both transcription and translation [3]. As a result, the reactions we consider for node are (17) modeling the following physical phenomena. We denote by protein decay, whereas represents the production rate that may be due to external inputs or perturbations (inducer, noise or disturbance). The second reversible reaction in (17) describes the binding of parent with multimerization factor to promoter complex forming complex, where and are the association and dissociation rate constants, respectively. Furthermore, each promoter complex will contribute to the production of through the production rate constant, modeled by the third reaction in (17). This production rate constant is a lumped parameter that incorporates features such as the RBS strength and the promoter strength. Finally, we assume that the total concentration of the promoter, denoted by, for each transcription component is conserved, so that, where is the number of possible complexes formed with the promoter of. This concentration is proportional to the concentration of copies of the promoter, which can be controlled, for example, by changing plasmid copy numbers in synthetic systems. The reaction flux vector contains all the reactions in the system, that is, binding/unbinding and protein production/decay. Given that binding/unbinding reactions occur on a much faster time-scale than protein production/decay [1], we partition into and, where contains the slow processes, whereas is composed of the fast reactions, that is, (18) Since the production of a typical protein takes approximately 5 minutes [1], and a few dozen mRNAs can be transcribed from the same gene simultaneously by [49], and similarly, a few dozen proteins can be translated from the same mRNA at the same time by [50], the effective production rate of protein from a gene can be as high as. This value can be arbitrarily decreased, for instance, by decreasing the RBS strength in synthetic circuits. The cell volume of Escherichia coli is typically between by [51], so that 1 molecule/cell corresponds to approximately concentration. By [52], a typical value of the dissociation constant of bacterial promoters is, whereas [22] suggests, and experimentally obtained values are provided in [53]. One of the most widely used high copy number vectors is the pUC plasmid [54], which can have hundreds of copies per cell [55]. A frequently used medium copy number plasmid is p15A with a few dozen copies per cell [56], whereas pSC101 is regarded as a low copy number plasmid with only a few copies per cell [56]. Finally, since the lifetime of a protein is on the order of a cell-cycle [1], we have [50]. The typical range of macroscopic parameters in Escherichia coli is summarized in Table 1. If we had not neglected mRNA dynamics, there would be three different time scales in the system. Binding and unbinding reactions occur on the time scale of seconds (or even subseconds) [1], representing the fastest time scale. The intermediate time scale is that of mRNA dynamics, as the average lifetime of mRNA is on the time scale of minutes [57], [58], [59]. Finally, the dynamics of proteins evolve on the slowest time scale (hours). As we are interested in describing the dynamics of the system on the time scale of gene expression, the concentration of promoter complexes and mRNA transcripts can be both approximated with their quasi-steady state values, leading to the models we have proposed in the paper. However, we would like to point out that including mRNA dynamics would not change anything substantial in the results and it would simply add more ODEs to the ODE model of an -node module without any effects on the retroactivity matrices (shown in [8] considering a specific example). First, note that in (2) has a block diagonal structure, yielding (19) where denotes the reaction flux vector corresponding to reversible binding reactions with the promoter of. Let denote the vector of concentrations of complexes with the promoter of at the quasi-steady state, obtained by setting. We first define as follows: (20) where is the entry in and is the number of complexes with the promoter of. Next, define the matrix as follows: it has as many columns as the number of complexes formed with the promoter of, and as many rows as the number of parents of: such that its element is if the parent of is bound as an -multimer in (if the parent is not bound). Finally, for nodes having parents, define the retroactivity of node as (21) For the most common binding types, and are given in Figure 3. For details on their derivation, see Supporting Text S3.
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.
systems biology
synthetic biology
biology
Modular Composition of Gene Transcription Networks