Network Inference, Analysis, and Modeling in Systems Biology

Cells use signaling and regulatory pathways connecting numerous constituents, such as DNA, RNA, proteins, and small molecules, to coordinate multiple functions, allowing them to adapt to changing environments. High-throughput experimental methods enable the measurement of expression levels for thousands of genes and the determination of thousands of protein–protein or protein–DNA interactions. It is increasingly recognized that theoretical methods, such as statistical inference, graph analysis, and dynamic modeling, are needed to make sense of this abundance of information. This perspective argues that theoretical methods and models are most useful if they lead to novel biological predictions and reviews biological predictions arising from three systems biology topics: graph inference (i.e., reconstructing the network of interactions among a set of biological entities), graph analysis (i.e., mining the information content of the network), and dynamic network modeling (i.e., connecting the interaction network to the dynamic behavior of the system). The methods and principles discussed in this perspective are generally applicable, and the examples were selected from plant biology wherever possible. To understand the function of a cell or of higher units of biological organization, often it is beneficial to conceptualize them as systems of interacting elements. For such a systems-level description (which represents the main goal of systems biology), one needs to know (1) the identity of the components that constitute the biological system; (2) the dynamic behavior of these components (i.e., how their abundance or activity changes over time in various conditions); and (3) the interactions among these components (Kitano, 2002). Ultimately, this information can be combined into a model that is not only consistent with current knowledge but provides new insights and predictions, such as the behavior of the system in conditions that were previously unexplored. The origins of systems biology can be traced back to systems theory, a line of inquiry based on the assumptions that all phenomena can be viewed as a web of relationships among elements, and all systems can be handled by a common set of methods (von Bertalanffy, 1968; Weinberg, 1975; Bogdanov, 1980; Heinrich and Schuster, 1996; Francois, 1999; Voit, 2000). Early attempts at systems-level understanding of biology suffered from inadequate data on which to base the theories and models; however, the recent advent of high-throughput technologies brought an abundance of data on system elements and interactions, leading to a revival of systems biology. Hypothetical Network Illustrating Network Analysis and Dynamic Modeling Terminology. (A) The interaction graph formed by nodes A to F consists of directed edges signifying positive regulation (denoted by terminal arrows), such as AB and ED, directed edges signifying negative regulation (denoted by terminal filled circles), such as FE and autoinhibitory (decay) edges (denoted by terminal filled circles) at nodes B to F. The graph contains one feed-forward loop (ABC; both A and B feed into C) and one negative feedback loop, EDF, which also forms the graph's strongly connected subgraph. The in-cluster of this subgraph contains the nodes A and B, while its out-cluster is the node C. (B) The node in-degrees (kin) that quantify the number of edges that end in a given node range between 0 (for node A) and 4 (for node C). The node out-degrees (kout), quantifying the number of edges that start at a given node, range between 1 (for C) and 3 (for B and E). The graph distance (d) between two nodes is defined as the number of edges in the shortest path between them. For example, the distance between nodes E and D is one, the distance between nodes D and E is two (along the DFE path), and the distance between nodes C and A is infinite because no path starting from C and ending in A exists. The betweenness centrality (b) of a node quantifies the number of shortest paths in which the node is an intermediary (not beginning or end) node. For example, the betweenness centrality of node A is zero because it is not contained in any shortest paths that do not start or end in A, and the betweenness centrality of node B is three because it is an intermediary in the ABD, ABDF, and ABDFE shortest paths. The in-(out-)degree distribution, [P(kin) and P(kout)] quantifies the fraction of nodes with in-degree kin (out-degree kout). For example, one node (C) has an out-degree of one; two nodes (E and F) have an out-degree of two and three nodes (A, B, and D) have an out-degree of three; the corresponding fractions are obtained by dividing by the total number of nodes (six). The distance distribution P(d) denotes the fraction of node pairs having the distance d. The betweenness centrality distribution P(b) quantifies the fraction of nodes with betweenness centrality b. (C) Hypothetical time courses for the state of each node in the network (denoted by SA to SE). The node states in this example can take any real value and vary continuously in time. The initial state (at t = 0) has state 1 for node A and 0 for every other node. Each node state approaches a steady state (a state that does not change in time), indicated in the last column (at t = ∞) of the time course. Network inference methods presented in section 2 use expression knowledge (such as the logarithm of relative expression with respect to a control state) such as this state time course to infer regulatory connections between nodes (i.e., the interaction network shown in [A]). State time courses like this also arise as outputs of continuous models. (D) The transfer functions of a hypothetical continuous deterministic model based on the interaction network (A) that leads to the time course under (C). Each transfer function indicates the time derivative (change in time) of the state of a node (denoted by a superscript ′ on the node state) as a function of the states of the nodes that are sources of edges that end in the node, including the node itself if it has an autoregulatory edge. The transfer functions in this hypothetical example are linear combinations of node states, with a positive sign for activating edges and negative sign for inhibitory edges, and all coefficients (parameters) are equal to unity. In general, transfer functions are nonlinear and have parameters spanning a wide range. (E) Hypothetical discrete time courses for each node in the network, where the node states can only take one of two values: 0 (off) and 1 (on). Discrete states such as this are obtained using a suitable threshold and classifying expression values as below threshold (0) and above threshold (1). The initial state (at t = 0) has state 1 for node A and state 0 for all other nodes. As in (C), each node reaches a steady state, indicated in the last column (at t = ∞) of the time course. Binary time courses such as this form the basis of Boolean network inference methods presented in section 2. State time courses like this also arise as outputs of Boolean models. (F) The transfer functions of a hypothetical Boolean model based on the interaction network (A) that leads to the time course in (E). Each transfer function indicates the state of the node at the next time instance (t + 1), denoted by a superscript asterisk on the node state, as a logical (Boolean) combination of the current (time t) states of the nodes that are sources of edges that end in the node. In this case, the autoinhibitory edges are not incorporated explicitly, assuming that positive regulation, when active, can overcome autoinhibition. Decay (switching off) after the positive regulators turn off is taken into account implicitly by not including the current state of the regulated node in its transfer function. It is assumed that the state of the node A does not change in time. The inhibitory edge FE is taken into account as a “not SF” clause in the transfer function of node E. More than one activating edge incident on the same node in general can be combined by either an “or” or “and” relationship, depending on whether they are closer to being independent (in case of “or”) or conditionally dependent or synergistic (in case of “and”). In this example, the edges AC and BC are assumed to be synergistic and independent of EC, and the edges BD and ED are assumed to be independent of each other. This essay focuses on the biological predictions arising from three related topics of importance in systems biology: graph inference, graph analysis, and dynamic network modeling. Graph inference refers to the problem in which the information on the identity and the state of a system's elements is used to infer interactions or functional relationships among these elements and to construct the interaction graph underlying the system. Graph analysis means the use of graph theory to analyze a known (complete or incomplete) interaction graph and to extract new biological insights and predictions from the results. Dynamic network modeling aims to describe how known interactions among defined elements determine the time course of the state of the elements, and of the whole system, under different conditions. A dynamic model that correctly captures experimentally behavior to the changes in the system's behavior to three of inquiry are often combined in the they three of the same to and if control a the dynamic behavior of biological interacting The predictions obtained from these methods range from of new interactions graph inference and of components and pathways graph analysis and dynamic network determination of parameters dynamic and of such as interaction or functional all three methods The most use of graph inference is using expression information to network (i.e., to which which other or A regulatory among two genes can be by experimental of the interactions and regulatory relationships among the two with expression in time or experimental conditions can be using such as the based on data from the to quantify selected or the data set methods into of genes that in a to conditions and that be however, that two nodes to the same does not a among them. The to extract from on the biological understanding of the that are biological by that they the function of by their with recognized analysis methods, such as analysis and the to the in the expression of a number of by the data into two or three that the between all the The is also to a by into independent and dependent the components of the dependent and independent and them by a linear and This used to the of in signaling in to of leading to the of cell to and of the of signaling A analysis with a number of to a expression data set previously genes that are in plant to experimental of the function of one of these genes presented by methods to a (i.e., feedback graph the relationships among components of a system and a set of that these relationships 2000). The starting edges are based on an initial of the experimental data and are by an the network and distribution the state of each node are inference used to infer the signaling network for cell to based on of signaling states different combinations of The network novel between and as as between and cell of Network The of the of to The network nodes genes and are by into genes genes and other genes The edges relationships in a between and corresponding to are by and inhibitory edges are by regulation is indicated by edge shown as filled or A filled to a line indicates where an edge starting from a regulatory node to nodes. A filled to a line indicates the case when nodes the same node. from from known information is by deterministic methods, such as analysis and or of and For example, a of changes in an model that were needed to the between model predictions of and experimentally data of experimental are as that indicates the of a or in a or to a in an where the expression or activity is is an example of such connecting the and can be by two paths of in the underlying interaction one connecting to and the other connecting the to inference relationships and interactions to the most network consistent with all experimental This used to the network corresponding to in in to the and is in the on the of interaction or regulatory relationships incorporated as edges of the biological interaction network have In interaction the nodes are proteins, and two are connected by a edge if is of their The of regulatory two node with and and has two of directed which to regulation (which can be positive or and 2002). have in various of two of the being the nodes are and edges in the same and the nodes are and edges at one and both interactions and and their edges are the of protein–protein and protein–DNA interactions protein–protein interactions, regulation, and expression or and regulation of genes The of high-throughput interaction and and of has to the of interaction for a number of In plant the interaction from the knowledge of interacting in and As in this graph analysis of the a of among their as as a however, that is a of in the number of for each interaction in the of these and in the of the interactions in the the predictions arising from network analysis as information The of interaction can be by network information content from of nodes or to whether all nodes are two are in as is by a of as a small number of can lead to wide The most network describe the among the importance of and the or of the network in of a given node A path of a from a to an end in a network or a of in a The distance between any two nodes in a network is defined to be the number of edges in the shortest path connecting nodes. the edges of a network are with the distance between two nodes be the of edge the path for which this is a The path of including and interaction and is than This that these are of to or also to path and the of multiple paths between a of nodes and This network to multiple between the same and and that these be to for in the In only a fraction of the nodes in the network be to any given node. The of nodes connected by paths in both and form the strongly connected can also the in-cluster that can the strongly connected but that be from and out-cluster of each of these to have a for example, in the nodes of the in-cluster to be in the nodes of the strongly connected form a signaling and the nodes of the out-cluster are for the of genes and for changes interaction including the have a strongly connected connecting the of the This a for of a or can the network and can have or the of do not have strongly connected a regulation with The and are with to of the nodes the strongly connected and This range of from regulatory to strongly connected interaction is by the that interactions are by edges is to the of current experimental as new information on the and of interactions leads to to of the edges, the of the strongly connected regulatory are than interaction and new to these their as are often regulated an regulatory network be a and have and feedback than a on regulation It be to whether new experimental and novel network the range of among interaction In to the the network of one also can interaction which are small (i.e., of the that have such as a negative feed-forward or of interactions have a higher than based on the subgraph of model to as analysis of the dynamic by and that dynamic to small in node states is with the relative of these to the that interaction form of and in by For example, the abundance of negative feedback in the of and of positive feedback at to or and to and The and of connections with a given node can be into of that centrality The such is the node or the number of edges to that node. the of interaction is a total can be into an in-degree and quantifying the number of and edges to the node The importance of any node in or the network is by its betweenness which is defined as the fraction of shortest paths between pairs of other nodes that node the node or betweenness centrality of a node is a this information can be into a description of the network by the distribution which the fraction of nodes in the network having A number of interaction including interaction and and regulatory a for node that the of a node that be used to the of the nodes in the with this are often to as in and and The distribution of is generally to a = where A is a and the is between 2 and from the with are also the in-degree distribution of and the distribution of have a small that regulation by is than regulation of by the same and that different are In nodes are most however, the nodes have that are of higher than the in general nodes are to as This leads to the that in node do not a of the of the the of the network into and 2002). This has experimentally in where the of a has shown to with the number of interactions in which the is a but of functional as are examples of nodes that are for and Ultimately, a of nodes to take into account the biological identity of the nodes and the synergistic and dynamic of the interactions and information than it for most interaction the state of the knowledge on these a suitable combination of node with betweenness and other centrality the between and The graph or combined with information the network nodes (such as the functional of the corresponding biological predictions on from interactions to functional The functions of can be on the basis of the of their interacting as it for and using and data interactions can be using based on the of interaction the network and functions and interactions can be between interaction in different interaction of two can be used to as a basis for in a recent analysis of and in of with the network a different than that by The connected of a network have used to connected The functional of genes each the of that are of the of an and network predictions of and expression changes in E. and The nodes of interaction of or other The of these can range from a of an or to or thousands of and they vary in time and in to or To these the interaction network needs to be by the state (i.e., or of each node and by a set of how the state of each node changes in to changes in the state of its In other the interaction network needs to be into a dynamic network Dynamic network models have as the interaction network, the transfer functions how the state of each node on the state of its and the initial state of each node in the system. of transfer functions for or functions for regulatory relationships and parameters values to be known or the model refers to such as based on the node states and transfer functions on and 2000). the interaction network, transfer functions, and initial states, the model the time of the state of the system. The most of a dynamic system is the number and of different often that are in the infinite time initial conditions that to a given constitute its of The of regulatory are to to states or while the of a network to steady state or to the of a given A dynamic model that correctly captures experimentally behavior to the changes in the system's behavior to to between and to conditions in which the of are It is to use a model to for that have a or beneficial on system behavior than it is to on the system; for example, models can multiple small that when the of using models are the information and data to construct a dynamic model are for all but the to a set of that are but be a model have a of in the interactions, and parameters it be to or it a of understanding or it be and its predictions be it be general to a number of and it be to small changes in parameters or Dynamic modeling are two continuous discrete and deterministic The refers to the of in the of the node state, while the indicates whether the transfer functions any or and are in biological a continuous model has the to describe the system; however, it also has the for information. A continuous deterministic the most used represents the of the corresponding continuous model as the number of or the to continuous deterministic models theoretical methods such as analysis that the analysis of where the system's changes as a function of various The of these can to the of the models. Discrete deterministic models a of in that they node states into a of expression or the this means that they and can be in where the number of continuous models or the the predictions of these models are and than the predictions of continuous models. deterministic models node states by and describe the of or of all components by based on and on a starting knowledge of the and the these models can and the of For example, a model of an signaling network for cell after of with to of the signaling two previously negative feedback and of the parameters the number of is methods can be used to parameters that account for dynamic For example, continuous deterministic modeling of the of combined with and analysis, to the of two new network nodes and of a novel of three feedback on the and the as a of one of the novel network nodes deterministic models of regulatory or signaling can also be with of cell and to in cell or For example, a recent model of plant by that the underlying is a feedback loop between relative in and It is that this feedback is the between and is regulated in such a that a higher in a cell leads to an at the that in a higher into that cell The of biological is taken into account by to Discrete (such as the of and for can be incorporated by the node states by the number of each and the time of the of each of a system's states and A recent model of the signaling and its in for signaling with a description of the states and This model the experimentally to different and that the in of from a Discrete Dynamic The of as a of in This Boolean model the of information in the of each in the states of signaling proteins, and in the of small by a number of that over relative and node initial The are as the of that the on state for the node In all with the of to with that in the the of in the of (A) The model that of or at the total of (B) The model that in circles), or lead to (C) The model that than than in the from dynamic models a Boolean description of regulation with continuous and by each node with both a continuous to a and a Boolean to and For example, a model of the regulation of the that its control is by a and a model of that of can have as as the of different dynamic models can be and the predictions by them are to the systems they is a of common arising from these models. For example, is that are from with not of of these to a (i.e., they a to changes in but their steady state is independent of the and from to at a value of the or in the these dynamic do not on the of the transfer functions or on the parameters and are by the underlying for example, positive feedback or a negative feedback often leads to and a negative feedback loop with a time The of general from the from models is an and in systems biology. biology an and feedback among and theoretical High-throughput or the inference the edges of interaction analysis of these into the of regulation, back to network inference and and biological Dynamic modeling of systems with and outputs the of regulatory components or of model predictions the or of the which in turn the to predictions and the of new biological Network analysis and dynamic network modeling approaches most for different network Network analysis can be on with of thousands of nodes and however, it the and of the corresponding to the edges of the deterministic or models for dynamic analysis of small but in for small in the number of nodes and edges and can be used on A is the of modeling that the of a network This perspective essay has shown a small of modeling in systems the is to and from different and To of systems biology has on which of As the of inquiry from to and as to to while to on to are to and the of between different of the same plant as as in these also that plant are at as as of and use components and can as model systems and in importance as the of systems biology of the on systems biology the that in for the to the of theory, and that be needed to understand and regulatory to to the examples shown in this perspective new and on plant systems biology in the is by and as as