Pattern storage in gene-protein networks

In this paper we study the potential of simple linear gene-protein interaction networks to store sparse input-output patterns. This problem bears similarity to the engineering task of reconstructing gene networks from time series of expression data, such as sets of microarrays. This is currently an intensively studied field where some results are directly applicable to our context. The central question in this study concerns the memory capacity of a network of n genes and proteins, which interact according to a simple linear state space model with p external outputs. Here it is assumed that to a certain combination of inputs u* there exists an optimal state of the system x*, i.e. values of the gene expressions and protein levels, that has been attained externally, e.g. through evolutionary learning. Given such a set of m learned optimal input-out patterns, the design question here is to find a sparse and hierarchical network structure for the gene-protein interaction matrix A, and the gene-input coupling matrix B. This problem is formulated as an optimization problem in a linear programming setting. From this formulation it is directly evident that the maximum number of patterns that can be stored is: mmax = n + p – 1. Furthermore, it allows for a numerical analysis, which shows that there are clear scale-invariant phase transitions for the sparsity, i.e. the relative number of zero-elements, in the matrices A and B as the number of patterns m increases. The sparsity in A and B exhibits continuous second-order phase transitions as the number of patterns reaches m1 = p/2 and m2 = 2p/3 respectively. These phase transitions divide the system in three regions with different memory characteristics. In the first region, below m = m1, the system stores patterns by directly connecting the inputs to the outputs, without directly involving the genes and proteins. In the second region, between m = m1 and m = m2, information is preferentially stored in matrix A, and in the third region, above m = m2, there is no clear preference for storing information in either A or B, and their sparsities behave increasingly identical. It is possible to formulate simple scaling rules for the behaviour of the sparsity in A and B versus m, though the exact morphology of these relations is not scale-invariant. Finally, numerical experiments are described that show that the patterns are stable within a certain finite range around the patterns.