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PLoS One. 2010; 5(10): e13080.
Published online 2010 October 8. doi:  10.1371/journal.pone.0013080
PMCID: PMC2951892

Using Effective Subnetworks to Predict Selected Properties of Gene Networks

Johannes Jaeger, Editor

Abstract

Background

Difficulties associated with implementing gene therapy are caused by the complexity of the underlying regulatory networks. The forms of interactions between the hundreds of genes, proteins, and metabolites in these networks are not known very accurately. An alternative approach is to limit consideration to genes on the network. Steady state measurements of these influence networks can be obtained from DNA microarray experiments. However, since they contain a large number of nodes, the computation of influence networks requires a prohibitively large set of microarray experiments. Furthermore, error estimates of the network make verifiable predictions impossible.

Methodology/Principal Findings

Here, we propose an alternative approach. Rather than attempting to derive an accurate model of the network, we ask what questions can be addressed using lower dimensional, highly simplified models. More importantly, is it possible to use such robust features in applications? We first identify a small group of genes that can be used to affect changes in other nodes of the network. The reduced effective empirical subnetwork (EES) can be computed using steady state measurements on a small number of genetically perturbed systems. We show that the EES can be used to make predictions on expression profiles of other mutants, and to compute how to implement pre-specified changes in the steady state of the underlying biological process. These assertions are verified in a synthetic influence network. We also use previously published experimental data to compute the EES associated with an oxygen deprivation network of E.coli, and use it to predict gene expression levels on a double mutant. The predictions are significantly different from the experimental results for less than An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e001.jpg of genes.

Conclusions/Significance

The constraints imposed by gene expression levels of mutants can be used to address a selected set of questions about a gene network.

Introduction

Living systems are typically able to maintain their physiological state under environmental changes and isolated genetic mutations [1]. This robustness, referred to as homeostasis [2] or canalization [3], [4], is achieved through feedback within highly connected regulatory networks of genes, proteins and metabolites [5][10]. For example, an action that reduces the expression of one gene may cause coordinate changes in other nodes to leave the physiological state unaffected. If a genetic mutation blocks one pathway, other avenues on the associated network may take its place. Unfortunately, this systemic stability often makes it difficult to eliminate defects in a biological network, as evidenced by the surprising lack of efficacy of many drugs that were designed to act on single molecular targets [11], [12]. The coupling can also lead to side effects from medications. For example, anti-inflammatory COX-2 inhibitors (e.g., Vioxx) cause adverse cardiovascular effects due to a concomitant imbalance of the lipids prostacyclin and thromboxane A2, which lie on the same network [13]. Clearly, the most effective and least detrimental changes in a biological process are implemented by altering the system in its entirety. This task requires predictive mathematical models which can be constructed from experimental data. In this paper, we propose an approach for such a construction.

There are hundreds of genes, proteins, and other molecular participants associated with most biological processes. Gene regulatory networks model all interactions between these nodes. However, the forms of these dependencies, as well as kinetic parameters such as reaction rates and diffusion constants are, at best, only known approximately [14]. It is unlikely that gene regulatory networks which are sufficiently accurate to make quantitative predictions on the underlying biological processes will be available in the near future [15], [16].

Many approaches to reduce the complexity of regulatory networks have been proposed [5]. Small modules or network motifs [17], [18] associated with specific tasks have been identified. Boolean variables [19] can reduce the complexity, although the coarse-graining will limit predictability to qualitative characteristics such as bifurcations. In gene influence networks [14], [20], a gene, its transcript, and protein are represented by a single node, which is quantified by the expression level of the mRNA. Regulatory interactions between nodes of an influence network include actions mediated by other components in the network.

Consider an influence network containing An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e002.jpg genes An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e003.jpg; denote the expression level of the An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e004.jpg gene by An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e005.jpg and the state of the network by An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e006.jpg. The influence network can be modeled by a set of ordinary differential equations An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e007.jpg

equation image
(1)

Steady states of influence networks can be obtained from DNA microarray experiments [5]. However, most influence networks contain hundreds of genes; thus, even if An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e009.jpg are assumed to have a simple (e.g., linear) form [21], a prohibitively large number of microarray experiments will need to be conducted in order to compute An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e010.jpg. Moreover, gene expression levels in microarray experiments have large (An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e011.jpg) error bars; when An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e012.jpg is large, the inversions needed to compute An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e013.jpg will exacerbate the uncertainty to a level which will make predictions difficult. One possibility is to only extract partial information on these networks through inference algorithms such as Network Identification by Multiple Regression [14], and Mode-of-action by Network Identification [22].

We propose an alternative approach. Rather than attempting to construct an accurate model of a gene network, we ask what questions on the network can be addressed (perhaps approximately) using low-dimensional and highly simplified effective models constructed from empirical data. What data would be needed for the construction? Will issues addressed through the approach be useful in applications?

We first note that genes in an influence network can be partitioned into strongly coupled subgroups or clusters. This partition can be made either using co-expression under genetic perturbations [23][25], or through the use of the Gene Ontology (GO) database (http://geneontology.org). Our main assumption is that the behavior of all nodes within a cluster can be controlled by imposing suitable changes in a small, specially chosen, subset of its members. The set could include genes that translate to transcription factors, and would hence influence many other genes [26], [27]. It may also include microRNAs within the cluster, each of which affect many genes through post-transcriptional regulation [28], [29], even though their fold induction on each gene is small.

Suppose we have partitioned the An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e014.jpg genes of the influence network into clusters, and identified a small number of genes/microRNAs from each cluster that can be used to control the expression levels of the remaining genes. Denote the set of these nodes by An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e015.jpg. The number An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e016.jpg of nodes in An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e017.jpg is much smaller than An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e018.jpg. We will represent their expression levels by An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e019.jpg, and re-index the variables so that the remaining expression levels are An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e020.jpg. With the new ordering, we write the state of the network as An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e021.jpg where we will refer to An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e022.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e023.jpg, as “internal” and “external” variables respectively.

In this paper, we limit consideration to networks with steady state solutions. When external perturbations are made on genes within An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e024.jpg, expression levels of the remaining genes at equilibrium are determined by Eqn. (1). These steady states lie on an An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e025.jpg-dimensional surface in An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e026.jpg, which we denote by An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e027.jpg. Figure 1(a) shows a schematic (2-dimensional) solution surface for the synthetic network introduced in the Results Section.

Figure 1
Example of an An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e028.jpg-dimensional solution surface An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e029.jpg of (1).

We make the following observations on solutions of the system An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e043.jpg. First, we assume that the unconstrained system has a unique stable steady state which we denote by An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e044.jpg. It satisfies the An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e045.jpg equations An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e046.jpg. The point An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e047.jpg representing it lies on An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e048.jpg. Next, consider the single knockout mutant An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e049.jpg (assumed to be viable) obtained by knocking out the An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e050.jpg gene. Since An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e051.jpg is set externally, the An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e052.jpg equation of (1) is no longer valid. The solution for the expression levels is obtained by solving the remaining An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e053.jpg equations. We denote this equilibrium by An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e054.jpg with An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e055.jpg, and represent it by An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e056.jpg. Since the equilibrium is associated with changes made within the set An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e057.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e058.jpg lies on An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e059.jpg. Consequently, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e060.jpg as well as An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e061.jpg for An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e062.jpg lie on An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e063.jpg, see Figure 1(b).

Our goal is to construct a system, referred to as the “effective empirical subnetwork” (EES), that can be computed using the gene expression levels of mutants discussed above, and whose equilibria are close to An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e064.jpg. Observe that An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e065.jpg and the An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e066.jpg points An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e067.jpg define a unique An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e068.jpg-dimensional plane in An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e069.jpg, which we denote by An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e070.jpg. Figure 1(b) shows the surface for the example above. The EES describes the set An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e071.jpg as parametrized by the internal variables. Since both An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e072.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e073.jpg contain the points An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e074.jpg, and An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e075.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e076.jpg), we expect them to be close in the region of interest.

Observe that the An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e077.jpg is a linear function determined by An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e078.jpg, and An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e079.jpg, (An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e080.jpg), but is otherwise independent of An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e081.jpg. In particular, each An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e082.jpg is a linear function of An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e083.jpg. Since An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e084.jpg lies on An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e085.jpg

equation image
(2)

for each An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e087.jpg. The coefficients An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e088.jpg can be evaluated by noting that An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e089.jpg lie on An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e090.jpg. There is one additional complication, that we illustrate using the following example. Suppose we consider a mutant where only An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e091.jpg is externally set. The remaining expression levels of the steady state of this mutant are solved using the last An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e092.jpg components of Eqn. (1). In particular, the expression levels An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e093.jpg of the internal variables in this mutant depend on An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e094.jpg. In general, the internal variables themselves depend on the gene expression levels whose values are externally imposed. Thus, we expect there to be relationships between the internal variables as well. As we show in the Methods Section, these dependencies can be assumed to take the form

equation image
(3)

for An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e096.jpg.

We have thus implemented two significant simplifications. The original system An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e097.jpg contained a large number (An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e098.jpgseveral hundred) of nonlinearly coupled variables. In contrast, the An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e099.jpg has a small number (An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e100.jpg) of internal variables, is linear, and can be constructed using the steady state solutions of the original system (wild-type) and An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e101.jpg single knockout mutants. Clearly, the EES is not an accurate representation of the original network. The issue is whether there are questions about An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e102.jpg that can be addressed using the EES. As we show below, this is indeed the case due to geometrical constraints imposed on the solution surface. Specifically, the EES can be used to predict, approximately, the expression levels of all nodes in An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e103.jpg, when external changes are made within An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e104.jpg; e.g., double knockout mutants. The validity of the EES construction can be tested by comparing its predictions with microarray data from such mutants. More significantly, the EES can be used to compute how the equilibrium of the system can be moved from its initial state An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e105.jpg to a pre-specified set of expression levels defined by a point An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e106.jpg, see Figure 2. Since An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e107.jpg will, in general, not lie on the solution surface, it cannot be reached through changes within An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e108.jpg. Instead, we can use the EES to compute An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e109.jpg, which is the closest point to An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e110.jpg on the plane An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e111.jpg, see Figure 2. Since the surfaces An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e112.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e113.jpg are close, changes imposed on the system by the external actions are expected to be close to those computed from the EES. Below, we verify this proximity in a synthetic influence network.

Figure 2
Moving the equilibrium from An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e114.jpg to An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e115.jpg by implementing changes within An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e116.jpg.

Results

A Synthetic Influence Network

In the model system, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e127.jpg is a linear combination of sigmoidal Hill functions; specifically,

equation image
(4)

where An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e129.jpg is the Hill function and the Hill index An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e130.jpg is chosen to be 2. The action of the An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e131.jpg gene on the An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e132.jpg one is characterized by parameters An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e133.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e134.jpg, which are assumed to be independent of the state An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e135.jpg of the system. The action is activating if An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e136.jpg and inhibiting if An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e137.jpg. The system is constructed so that An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e138.jpg is a steady state of Eqns. (1). Numerically, we find that model systems defined by Eqns. (1) and (4), have at most one stable equilibrium. We suspect that this is due to restrictions imposed by the fact that the sign of each partial derivative An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e139.jpg is independent of the state of the system.

In order to compute the solutions to the knockout mutant An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e140.jpg, we set An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e141.jpg, and solve the remaining equations of (1) as a nonlinear least squares problem. When the normalized residue fails to fall below An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e142.jpg, it is assumed that the corresponding solution does not exist.

We report on a model system containing 21 nodes and shown schematically in Figure 3. We start with the three subnetworks, each of size 7. The vector An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e143.jpg for each of these subgroups consists of random entries between 0.5 and 1.5, and the matrix An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e144.jpg contains random values between 0 and 2. The entries of the Jacobian of the system given by Eqns. (1) and (4) at An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e145.jpg are An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e146.jpg; thus An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e147.jpg can be computed for a given set of An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e148.jpg's. Since we require An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e149.jpg to be stable, we insist that all eigenvalues of the Jacobian be negative. This is guaranteed by starting with a diagonal matrix with negative entries and performing a (random) orthonormal transformation. Once three such subnetworks are computed, their nodes are coupled by sparse, weak interactions. Each node in a subnetwork is coupled to only one from each of the other subnetworks, and the mean coupling strength is chosen to be 0.1 of the average coupling within subgroups.

Figure 3
A schematic of the synthetic network.

The EES is to be constructed using the expression levels of single knockout mutants. As illustrated in Figure 3, mutants An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e151.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e152.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e153.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e154.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e155.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e156.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e157.jpg, and An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e158.jpg in our example are not viable; i.e., when the corresponding An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e159.jpg is set to zero, the system (1) does not have a solution. The subset on which to construct the EES can contain any of the other nodes. In the work reported here, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e160.jpg (genes marked in red in Figure 3). The variables An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e161.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e162.jpg are re-indexed as described before. The An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e163.jpg is computed using the expression levels of all 21 genes at An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e164.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e165.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e166.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e167.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e168.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e169.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e170.jpg.

In order to illustrate the proximity of An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e171.jpg to An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e172.jpg, we use the following example, see Figure 1. Since we need to reduce the dimensionality for visualization, we fix the expression levels of (the re-indexed genes) An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e173.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e174.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e175.jpg, and An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e176.jpg at their values at An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e177.jpg; for our model, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e178.jpg. For each pair of values for An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e179.jpg, we solve the model system (1) for the remaining 15 expression levels. These solutions lie on 2-dimensional surface in An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e180.jpg. The gray surface of Figure 1 is An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e181.jpg. The 2-dimensional plane An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e182.jpg of the EES contains points An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e183.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e184.jpg, and An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e185.jpg.

Next, we compare expression levels of double knockout mutants predicted by the EES with the corresponding solutions of the model system (1). The 14 viable double knockout mutants of the system are An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e186.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e187.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e188.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e189.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e190.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e191.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e192.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e193.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e194.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e195.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e196.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e197.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e198.jpg, and An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e199.jpg. In each case, the expression levels of the 4 remaining nodes in An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e200.jpg, and the 15 nodes outside of An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e201.jpg are compared. We differentiate between these two groups.

Results for the first group (genes in An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e202.jpg) are as follows. Of the 56 comparisons, 46 EES predictions are within An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e203.jpg of the expression levels computed from (1), 3 others are between An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e204.jpg, and 3 between An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e205.jpg. Results for the second group (genes outside of An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e206.jpg) are as follows. Of the 210 expression levels to be compared, 170 EES predictions are within An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e207.jpg of the expression levels computed from (1), 30 more are between An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e208.jpg, and 7 others are between An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e209.jpg.

We finally demonstrate how the equilibrium of the system can be moved (near) to a pre-specified set of expression levels. The original equilibrium of our example is An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e210.jpg, with

equation image

We want to find out how the expression levels of genes in An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e212.jpg need to be changed so that the system moves to, or as close as possible to, a pre-specified set of expression levels for all genes. As an example, we attempt to change the equilibrium of the system to An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e213.jpg (see Figure 2) given by An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e214.jpg, where

equation image

Since we have computed the EES, we can calculate the projection An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e216.jpg of An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e217.jpg on An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e218.jpg. It is given by An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e219.jpg, where

equation image

Finally, we use the model system Eqns. (1) and (4) to compute the external variables when internal variables are fixed at An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e221.jpg. It is found to be An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e222.jpg, where An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e223.jpg, and

equation image

The Euclidean distances between the points are An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e225.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e226.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e227.jpg, and An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e228.jpg. Thus, we attempted to move the equilibrium from An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e229.jpg to a point An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e230.jpg that was a distance 0.55 away, but were only able to move it on An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e231.jpg to a point An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e232.jpg, which is a distance 0.40 away from An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e233.jpg. However, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e234.jpg is only a distance 0.15 from the point An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e235.jpg, which is the solution of the original system when expression levels of the internal variables are set to An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e236.jpg. We have found that An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e237.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e238.jpg are close in studies of several model systems and for many points An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e239.jpg. Thus, the EES can be used to pre-determine, approximately, the equilibrium of the original network when changes made within An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e240.jpg.

Transcriptional Regulatory Network in E.coli

The EES can be constructed using microarray data from the wildtype and single knockout mutants of genes in An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e241.jpg. It can then be used to predict gene expression levels of other mutants. This observation is of interest due to the availability of previously published data on an oxygen deprivation network in E.coli [30], [31]. Ref. [32] reports gene expression levels in the wildtype and in single knockout mutants of key transcriptional regulators in the oxygen response, namely An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e242.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e243.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e244.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e245.jpg, and An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e246.jpg, as well as in the double knockout mutant An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e247.jpg, in aerobic and anaerobic glucose minimal medium conditions. Since the oxygen deprivation network is not fully active under aerobic conditions, we focus on the behavior of E.coli under anaerobic conditions.

It should be noted that gene expression levels in E.coli are unlikely to be in a steady state; rather, the expression levels reported in Ref. [32] are averages from a group of cells in various stages in the cell cycle. The analysis in this Section assumes that the computation of the EES and its predictions are valid for these averages. Preliminary results from our current work on systems exhibiting circadian rhythms validate this assumption.

We construct An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e248.jpg as follows. In the Gene Ontology classification assigned by Affymetrix, the five genes An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e249.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e250.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e251.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e252.jpg, and An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e253.jpg have a common term “GO:0006355, Regulation of transcription, DNA-dependent.” Moreover, this is the only common classification for the five genes. We choose An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e254.jpg to be the set of all genes carrying this term. The full list of 299 genes is given in Supporting Information S1.

The data set GSE1121 of the GEO site (www.ncbi.nlm.nih.gov) [32] provides gene expression levels for four replicates of the wildtype and three each for the mutants. The replicates are used to estimate the mean and standard deviation for the expression levels of each gene in An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e255.jpg, see Supporting Information S1. Since the EES is linear, we rescale the expression levels of each gene by its (mean) value in the wildtype. Table 1 gives these rescaled expression levels for the internal variables An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e256.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e257.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e258.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e259.jpg, and An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e260.jpg under anaerobic glucose minimal medium conditions.

Table 1
Normalized gene expression levels in the wildtype and mutants.

Note that error estimates for the expression levels of several genes is large. This is the reason that a reduced network is essential in order to make useful predictions. Second, as seen from Table 1, reported expression levels of the gene An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e314.jpg in the mutant An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e315.jpg is non-zero. Presumably, what is measured are non-functional analogs of the corresponding genes. In calculating the EES, we set these expression levels (shown in parentheses in Table 1) to zero.

The component of the EES for the internal variables is

equation image
(5)

The next step is to compute the EES predictions for An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e317.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e318.jpg, and An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e319.jpg in the double knockout An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e320.jpg. This is done using the matrix (5) and setting An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e321.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e322.jpg to zero. Expression levels of the remaining genes in An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e323.jpg, predicted using the EES, are An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e324.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e325.jpg, and An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e326.jpg. We need to determine, at a 5% level of confidence, if these predicted values are consistent with those from the replicates of the double mutant. The comparison is made using the An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e327.jpg-test (ttest in MATLAB, The Mathworks, Inc.), and it is found that the null hypothesis, that experimental data comes from a (normal) distribution with mean equal to the computed gene expression level, is rejected at the 5% level only for appY.

Next, we implement the analysis for genes outside of An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e328.jpg. The null hypothesis cannot be rejected at the An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e329.jpg level for An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e330.jpg of the An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e331.jpg genes. The three experimental values of the expression level of each gene in the double knockout, the corresponding predictions of the EES, and the test statistic An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e332.jpg are given in Supporting Information S1. Since the Student's distribution associated with the comparison has two degrees of freedom, the null hypothesis is rejected when An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e333.jpg. The histogram of the test statistic for the 299 genes is shown in Figure 4(a). In Supporting Information S1, we highlight the genes for which the null hypothesis is rejected. We emphasize that, unlike in many prior studies whose assertions are limited to whether genes in mutants are up/down regulated, our predictions are quantitative.

Figure 4
Comparison of EES predictions with experimental gene expression levels of An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e334.jpg.

The proximity of the predicted and experimental values is not due to a lack of variability in the expression levels of genes in An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e335.jpg. We verify this by computing the differential expression of genes in the mutants. Figure 4(b) shows the histogram of the largest deviations from the wildtype, normalized by the standard deviation (between replicates) in the wildtype. Expression levels of over half the genes deviate by more than 2 standard deviations.

Discussion

An accurate model of the gene regulatory network associated with a hereditary disease can be used to compute the most effective and least detrimental treatment to prevent its onset. Unfortunately these networks contain hundreds of genes, proteins, and other molecules whose interactions are only partially known [5], [8][10]. It is unlikely that detailed models of such networks will be available in the near future. The question raised in the paper is whether information needed to move the steady state of a network can be deduced from an analysis of highly simplified, empirically determined models. The data used for analysis is obtained from microarray experiments.

Our approach is as follows. We first identify a (relatively) small set An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e336.jpg of An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e337.jpg nodes (internal variables) in the influence network which can be used to affect the remaining genes. (Mathematically, for each external variable An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e338.jpg, we require one or more of An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e339.jpg, where An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e340.jpg are the internal variables, to be non-vanishing.) Next, we limit consideration only to steady states of the network as internal variables are modified. Finally, this solution surface An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e341.jpg is approximated using the unique An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e342.jpg-dimensional plane An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e343.jpg defined by the gene expression levels of the wildtype and the An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e344.jpg single knockout mutants in An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e345.jpg; the model system whose solutions lie on the plane is the EES.

Some genes may be critical in the sense that the organism may not be viable when they are knocked out. There were similar nodes (shown in black in Figure 3) in our synthetic model. In our approach they cannot be used as internal nodes. However, if they need to be utilized, the EES can be computed using heterozygous mutants or those where the expression level is up/down regulated to a value other than zero through, for example, transfection [33], [34].

We emphasize that, due to the reduced dimensionality and its linearity, we do not expect the EES to be an accurate model of the original system. However, because of the geometrical constraints, it is possible to use the EES to (approximately) compute answers to a very limited set of questions about the system. Specifically, they are questions on gene expression levels when external changes are made within An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e346.jpg. As an example, the EES can predict gene expression levels in double knockout mutants. We tested the predictions using previously published data on a double knockout mutant in an oxygen deprivation network of E.coli. (Here, as in most cases, the underlying network is unknown.) We identified the group of 299 genes to be studied using the Gene Ontology database. The EES was computed using the expression levels of five single knockout mutants, and used to predict their expression levels in the double mutant. The predictions were significantly different from the experimentally obtained expression levels for less than 30% of genes.

Interestingly, the EES can be used to compute how expression levels of genes within An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e347.jpg need to be changed so that the equilibrium of the entire network is moved from its initial state An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e348.jpg to, or as close as possible to, a pre-specified position An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e349.jpg. We showed through an example that the solution computed using the EES is close to that of the full network. However, the efficacy of the move depends on the proximity of An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e350.jpg to the surface An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e351.jpg. If An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e352.jpg is far from An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e353.jpg, then the set of internal variables need to be expanded in order to find acceptable solutions.

Before concluding, we briefly address a few issues; the first is the observation that, in the parameter range considered, the model system given by Eqns. (1) and (4) have at most one stable steady state. Even though we required An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e354.jpg to be stable (by an appropriate choice of eigenvalues of the linearization), non-linear systems can, in general, be expected to have additional solutions. However, our model has a special feature: the signs of the partial derivatives An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e355.jpg are independent of the state of the system. The analogous biological statement is that, if nodes An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e356.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e357.jpg are isolated, the action of node An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e358.jpg on node An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e359.jpg increases in magnitude as An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e360.jpg increases. Is this condition, combined with the choice of eigenvalues, sufficient to guarantee a uniques stable solution? We are currently studying this question. It should be noted that the uniqueness of solutions has been proven for several other classes of monotonic nonlinear systems [35][37].

The second issue involves the partitioning of genes into clusters and the choice of internal variables. Internal variables in the oxygen deprivation network of E.coli were already determined from the experiments reported in Ref. [32]. We used the GO classification to identify nodes belonging to the network. Different approaches can be used to partition genes into clusters when biological classifications are not available. For example, one could use topological (e.g., persistent homology [38], [39]) or graph theoretic (e.g., spectral clustering [23], community clustering [24]) methods. Integrated genomic analysis, which successfully identified subtypes of gliobastoma [40], can also be used in clustering genes through the use of heat maps [41], [42]. The choice of internal variables requires biological input. Mathematically, the requirement is that each node in the cluster can be affected by suitable changes in internal variables. As we mentioned, genes that translate to transcription factors, or microRNAs [28], [29] within the cluster, could act as internal nodes.

Third, can one estimate the proximity of An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e361.jpg to the solution surface An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e362.jpg? Differences in gene expression levels of double knockout mutants are one measure of the proximity. Alternatively, we could use the corresponding differences in heterozygous single knockouts (whose expression levels are roughly half of the wildtype) and the predictions of the EES.

We believe that approaches similar to those outlined here can prove useful in treating complex genetic diseases by helping identify optimal combinations of up/down regulation of genes (or optimal combinations of single target drugs) that have minimal side effects and are most effective in moving the equilibrium of the network in its entirety to a preferred state. We hope our work motivates studies on this issue.

Methods

Construction of the EES

As illustrated in Figures 1, the EES is constructed so that, as internal variables are modified, the solutions of the system lie on the An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e363.jpg-dimensional plane An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e364.jpg. Thus the external variables are linear in An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e365.jpg's, and consequently, have the form given by Eqns. (2). We need to compute the coefficients An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e366.jpg for An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e367.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e368.jpg. This is done by noting that the expression levels An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e369.jpg of each of the An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e370.jpg mutants An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e371.jpg satisfies Eqn. (2), thus providing the conditions necessary to compute An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e372.jpg's.

We note, however, that the internal variables themselves are inter-related. For example, in the single knockout mutant An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e373.jpg, all expression levels (other than An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e374.jpg) are determined by solving the last An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e375.jpg equations of (1). Thus, we need to derive relationships between the internal variables. Consider for example, the dependence of An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e376.jpg on the remaining internal variables. In order to find its form, let us reduce the set of internal variables to An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e377.jpg; An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e378.jpg is now an external variable. Hence, with the approximations used in the paper, An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e379.jpg is a linear combination of the remaining internal variables. Since An external file that holds a picture, illustration, etc.
Object name is pone.0013080.e380.jpg is one solution of the system

equation image
(6)

Similar relationships are obtained for the other internal variables.

Supporting Information

Supporting Information S1

Tables showing (1) The set G of 299 genes chosen to study the oxygen deprivation network of E.coli. These genes have the common biological function 0006355 “regulation of transcription, DNA-dependent” (2) Mean values of the 299 genes in the wildtype and the mutants. (3) Standard deviation of 299 genes in the wildtype and mutants. (4) The coefficients of the Effective Empirical Network. (5) Comparison between the predicted and experimental gene expression levels for the double knockout of fnr and arcA. The experimental data are normalized by the corresponding mean value of the wildtype replicates (item (2)).

(0.20 MB XLS)

Acknowledgments

The authors would like to thank Tim Cooper, Chad Creighton, John Miller, and Gregg Roman for discussions.

Footnotes

Competing Interests: The authors have declared that no competing interests exist.

Funding: This work was partially funded by grant 0607345 from the National Science Foundation. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

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