Promotion/Inhibition Effects in Networks:
A Model with Negative Probabilities

Networks are often used to encode interactions between chemical or biological compounds, such as proteins and genes [1, 2, 3, 4, 5]. A distinguishing feature of such networks is that edge-weights quantify activation/suppression rates, equivalently, promotion/inhibition relations between nodes. These in turn impact expression levels of substances assigned to node sites. Measuring precisely the rates that regulate such interactions is often a difficult task, whereas measuring expression levels at the node sites is substantially easier. Thus, the subject of the present work is to develop a framework for solving the inverse problem to identify such network parameters (edge-weights) from information on expression levels and affinity between nodes. Specifically, we seek to determine sign-indefinite values for edge-weights of a network based on knowledge of nodal mass and the sign of the edge-weights. The sign of edge-weights is made available in the form of a sign-indefinite adjacency matrix. It is also possible that an estimate of (signed) edge-weights is already available and needs to be updated, so as to restore consistency with measured mass at the nodes. This situation is completely analogous to the case where the prior is only the signed adjacency matrix, and can be treated similarly.

We introduce a natural framework to identify sign-indefinite edge-weights from this type information, by adopting a sign-indefinite transition-probability model to encapsulate “promotion vs. inhibition.” Incorporating “negative probabilities” in a model was advocated early on by Paul Dirac [6] and Richard Feynman [7], and in some scarce work that followed [8, 9]. Dirac and Feynman argued specifically that negative probabilities, when not directly and experimentally measurable, are perfectly acceptable as reflecting internal unobservable manifestations of an underlying mechanism. In our setting however, the interpretation of negative transition probabilities as rates is physically meaningful. Activation/suppression rates, reflected in sign-indefinite edge-weights, dictate the stationary distribution at the nodes (e.g., protein levels, and so on) and are sought to explain the observed data.

The significance of identifying edge-weights in a gene regulatory network is of vital importance in that it allows assessing the role of nodes and edges in the overall functionality and robustness of networks [2, 3]. Indeed, chemical deactivation of sites in a gene regulatory network that may contribute to selective cell-death has been the rationale behind many types of medical treatment of cancer, and the endeavour to properly quantify functionality and robustness of gene networks with suitable metrics, that point to vital nodes and links, is on-going [10]. In particular, various centrality measures as well as network curvature have been proposed to quantify significance of nodes/edges as well as resilience of the network to changes. Determining the value of such metrics presupposes knowledge of edge-weights, which is the modest goal of the inverse problem that we consider herein.

A contribution of the present work is to propose identifying edge weights by minimizing a suitable relative entropy functional between sign-indefinite measures. The problem can be solved by a Sinkhorn-like iteration to obtain the minimizer. This new algorithm reduces to the well-known Sinkhorn “diagonal-scaling iteration” when the network adjacency matrix is sign definite. In the generality of the present model, with sign-indefinite edge weights, the algorithm amounts to an iterative scaling scheme with the scaling of multiplicative or inverse-multiplicative nature, depending on the sign of a corresponding entry in the adjacency matrix. At each step of the iteration, the scaling factor is readily available as the positive root of a quadratic polynomial. The new algorithm, much like the Sinkhorn iteration, can be seen as effecting coordinate ascent to obtain the maximum of a concave functional, displaying as a consequence linear convergence rates.

Below, in Section II we present our problem formulation, followed by theoretical development and algorithmic considerations that are presented in Section III. Section IV provides illustrative examples.

The data for our mathematical problem consist of two entities: first a symmetric matrix $A=A^{\prime}\in\mathbb{\{}-1,0,1\}^{n\times n}$, and then a (column) probability vector $p\in\mathbb{R}^{n}$, i.e., such that $p_{i}\geq 0$ for $i\in\{1,\ldots,n\}$ and $\sum_{i=1:n}p_{i}=1$.

As noted in the introduction, the matrix $A$ may represent the sign-indefinite adjacency of a given gene regulatory network. In such an example, the nodes $v\in\mathcal{V}=\{1,2,\ldots,n\}$ correspond to genes and the vertex set cardinality $|\mathcal{V}|=n$ is typically very large, with $n$ of the order of $10^{3}-10^{4}$. The adjacency matrix $A=[A_{ij}]_{i,j=1:n}$ encodes both, a “connectivity” structure among genes as well as a constructive/indifferent/destructive (promoting/no effect/inhibiting) contribution of the corresponding pair of genes to the respective protein production levels. This information is encoded in the sign of the respective entry $A_{ij}\in\{-1,0,1\}$, with positive value indicating a constructive co-expression, zero indicating no perceived effect, and negative indicating inhibiting effect. This (biological) information is assumed given to us.

The second piece of datum for our problem is a probability vector $p_{i}$ ($i\in\{1,\ldots,n\}$ that represents corresponding relative protein levels, and hence respective potency of the genes sites. That is, $p_{i}\geq 0$ and $\sum_{i=1:n}p_{i}=1$, with high values representing strong expression of respective proteins. Once again, this information is given to us from experimental data.

Biologists are typically interested in modeling the contributions of various genes on relative protein production levels in a quantitative manner. To this end, ad-hoc schemes have been utilized; see e.g., [3, 1] and the references therein. In earlier work, the rationale behind the proposed schemes was to employ the theory of Markov chains by calibrating the constructive/inhibiting effect of sites to production levels to only positive values. This was done by adding a suitable positive constant to all entries of $A$. Accordingly, $A$ becomes a positive matrix, to which the standard theory of Markov chains may be adapted, after scaling $A$ suitably, so that $p$ may be regarded as the stationary distribution of a corresponding Markov chain. Several pitfalls plague all such schemes, and are traceable to the difference between constructive and inhibiting effects, since adding a constant, evidently creates a bias in one direction and not the other.

In the present work, we propose an approach that seems natural for the problem at hand, and brings in the concept of negative probabilities. As indicated earlier, such concepts have had notable proponents including Dirac and Feynman, and some scant following. In some detail, we hereby, postulate and seek a sign-indefinite Markov transition model to explain the observed invariant distribution in $p$, while acknowledging the promoting/inhibiting nature of the links between genes. Feynman’s dictum suggests that, as long as internal probabilities in a model are not experimentally observable (as is the case with the sought transition kernel that will be constructed to respect the signs of $A_{ij}$’s), it is acceptable provided that it explains observed and experimentally measurable (non-negative) probabilities and relative frequencies.

Thus, we seek to identify a sign-indefinite probability transition matrix $\Pi=[\Pi_{ij}]_{i,j=1:n}$ such that $\Pi_{ij}\gtreqqless 0$ in accordance with the similar property for the corresponding $A_{ij}$. Alternatively, $\frac{\Pi_{ij}}{A_{ij}}\geq 0$, while $A_{ij}=0\Rightarrow\Pi_{ij}=0$. To this end, we formulate the following problem seeking to minimize a suitable functional (herein, entropy).

The use of relative entropy between $A$ and $\Pi$, as above, echoes a similar usage in the justifications of Schrödinger’s bridges, rooted in large deviations theory (see [11]. It has also been central in other problems in network theory [12]. Negative transition probabilities in our context can be interpreted as promotion/inhibition rates.

We first reformulate Problem 1 in the following manner. We define

This is seen to be a sign-definite adjacency of the network with entries $\mathbf{A}_{i,j}\in\{0,1\}$. Likewise, for a sign-indefinite transition probability $\Pi$ as earlier, we write

for the corresponding (typically) unnormalized transition probability matrix. That is, $\mathbf{\Pi}$ has non-negative entries, but without guaranteed row-sums being equal to one. Thus, our problem can be re-cast as follows.

It is clear that Problems 1 and 2 are equivalent, and of course, that $\mathbf{J}(\mathbf{\Pi},\mathbf{A})=J(\Pi,A)$. Problem 2 is quite similar to the classical Schrödinger problem [11], but because $A$ is sign-indefinite, (2b-2c) are not the usual conditions. Yet, they are still linear, and the above re-formulation readily leads to the following conclusion.

We now discuss representative examples to illustrate what can be accomplished with the proposed formulation. The code used for working out all of the examples may be found at https://github.com/dytroshut/negative-probability-forward-backward.

The purpose of the present work has been to introduce a sign-indefinite probabilistic model to reflect promoting/inhibiting affinity between nodes, as in co-expression of genes in gene regulatory networks. The theory is of independent interest and leads to a nonstandard extension of the classical Sinkhorn algorithm. An important question that remains is on how to efficiently test feasibility of Problems 1 and 2. In addition, in light of the fact that the Sinkhorn iteration addresses a static version of the more general Schrödinger bridge problem [11], it is of interest to explore a setting that allows modeling evolution of nodal mass, allowing for promotion/inhibition effects, under suitable dynamics.