I Pérez Castillo

Identifying essential genes in E. coli from a metabolic optimization principle

Understanding the organization of reaction fluxes in cellular metabolism from the stoichiometry and the topology of the underlying biochemical network is a central issue in systems biology. In this task, it is important to devise reasonable approximation schemes that rely on the stoichiometric data only, because full-scale kinetic approaches are computationally affordable only for small networks (e.g., red blood cells, ≈50 reactions). Methods commonly used are based on finding the stationary flux configurations that satisfy mass-balance conditions for metabolites, often coupling them to local optimization rules (e.g., maximization of biomass production) to reduce the size of the solution space to a single point. Such methods have been widely applied and have proven able to reproduce experimental findings for relatively simple organisms in specific conditions. Here, we define and study a constraint-based model of cellular metabolism where neither mass balance nor flux stationarity are postulated and where the relevant flux configurations optimize the global growth of the system. In the case of Escherichia coli, steady flux states are recovered as solutions, although mass-balance conditions are violated for some metabolites, implying a nonzero net production of the latter. Such solutions furthermore turn out to provide the correct statistics of fluxes for the bacterium E. coli in different environments and compare well with the available experimental evidence on individual fluxes. Conserved metabolic pools play a key role in determining growth rate and flux variability. Finally, we are able to connect phenomenological gene essentiality with “frozen” fluxes (i.e., fluxes with smaller allowed variability) in E. coli metabolism.

Analytic solution of attractor neural networks on scale-free graphs

We study the influence of network topology on retrieval properties of recurrent neural networks, using replica techniques for dilute systems. The theory is presented for a network with an arbitrary degree distribution p(k) and applied to power-law distributions p(k) ∼ k −γ , i.e. to neural networks on scale-free graphs. A bifurcation analysis identifies phase boundaries between the paramagnetic phase and either a retrieval phase or a spin-glass phase. Using a population dynamics algorithm, the retrieval overlap and spin-glass order parameters may be calculated throughout the phase diagram. It is shown that there is an enhancement of the retrieval properties compared with a Poissonian random graph. We compare our findings with simulations.

Parallel dynamics of disordered Ising spin systems on finitely connected random graphs

We study the dynamics of bond-disordered Ising spin systems on random graphs with finite connectivity, using generating functional analysis. Rather than disorder-averaged correlation and response functions (as for fully connected systems), the dynamic order parameter is here a measure which represents the disorder-averaged single-spin path probabilities, given external perturbation field paths. In the limit of completely asymmetric graphs, our macroscopic laws close already in terms of the single-spin path probabilities at zero external field. For the general case of arbitrary graph symmetry, we calculate the first few time steps of the dynamics exactly, and we work out (numerical and analytical) procedures for constructing approximate stationary solutions of our equations. Simulation results support our theoretical predictions.

The Little-Hopfield model on a sparse random graph

We study the Hopfield model on a random graph in scaling regimes where the average number of connections per neuron is a finite number and the spin dynamics is governed by a synchronous execution of the microscopic update rule (Little–Hopfield model). We solve this model within replica symmetry, and by using bifurcation analysis we prove that the spin-glass/paramagnetic and the retrieval/paramagnetic transition lines of our phase diagram are identical to those of sequential dynamics. The first-order retrieval/spin-glass transition line follows by direct evaluation of our observables using population dynamics. Within the accuracy of numerical precision and for sufficiently small values of the connectivity parameter we find that this line coincides with the corresponding sequential one. Comparison with simulation experiments shows excellent agreement.

Dynamical Replica Analysis of Disordered Ising Spin Systems on Finitely Connected Random Graphs

We study the dynamics of macroscopic observables such as the magnetization and the energy per degree of freedom in Ising spin models on random graphs of finite connectivity, with random bonds and/or heterogeneous degree distributions. To do so, we generalize existing versions of dynamical replica theory and cavity field techniques to systems with strongly disordered and locally treelike interactions. We illustrate our results via application to, e.g., +/-J spin glasses on random graphs and of the overlap in finite connectivity Sourlas codes. All results are tested against Monte Carlo simulations.

Finitely connected vector spin systems with random matrix interactions

We use finite connectivity equilibrium replica theory to solve models of finitely connected unit-length vectorial spins, with random pair-interactions which are of the orthogonal matrix type. Finitely connected spin models, although still of a mean-field nature, can be regarded as a convenient level of description in between fully connected and finite-dimensional ones. Since the spins are continuous and the connectivity c remains finite in the thermodynamic limit, the replica-symmetric order parameter is a functional. The general theory is developed for arbitrary values of the dimension d of the spins, and arbitrary choices of the ensemble of random orthogonal matrices. We calculate phase diagrams and the values of moments of the order parameter explicitly for d = 2 (finitely connected XY spins with random chiral interactions) and for d = 3 (finitely connected classical Heisenberg spins with random chiral interactions). Numerical simulations are shown to support our predictions quite satisfactorily.

A spherical Hopfield model

A spherical Hopfield-type neural network is introduced, involving neurons and patterns that are continuous variables. Both the thermodynamics and dynamics of this model are studied. In order to have a retrieval phase a quartic term is added to the Hamiltonian. The thermodynamics of the model is exactly solvable and the results are replica symmetric. A Langevin dynamics leads to a closed set of equations for the order parameters and effective correlation and response function typical for neural networks. The stationary limit corresponds to the thermodynamic results. Numerical calculations illustrate these findings.

On the strategy frequency problem in batch Minority Games

Ergodic stationary states of Minority Games with S strategies per agent can be characterized in terms of the asymptotic probabilities a with which an agent uses a of his strategies. We propose here a simple and general method to calculate these quantities in batch canonical and grand-canonical models. Known analytic theories are easily recovered as limiting cases and, as a further application, the strategy frequency problem for the batch grand-canonical Minority Game with S = 2 is solved. The generalization of these ideas to multi-asset models is also presented. Though similarly based on response function techniques, our approach is an alternative to the one recently employed by Shayeghi and Coolen for canonical batch Minority Games with arbitrary number of strategies.

Optimal capacity of the Blume-Emery-Griffiths perceptron

A Blume-Emery-Griffiths perceptron model is introduced and its optimal capacity is calculated within the replica-symmetric Gardner approach, as a function of the pattern activity and the embedding stability parameter. The stability of the replica-symmetric approximation is studied via the analog of the de Almeida-Thouless line. A comparison is made with other three-state perceptrons.

Gardner optimal capacity of the diluted Blume–Emery–Griffiths neural network

The optimal capacity of a diluted Blume–Emery–Griffiths neural network is studied as a function of the pattern activity and the embedding stability using the Gardner entropy approach. Annealed dilution is considered, cutting some of the couplings referring to the ternary patterns themselves and some of the couplings related to the active patterns, both simultaneously (synchronous dilution) or independently (asynchronous dilution). Through the de Almeida–Thouless criterion it is found that the replica-symmetric solution is locally unstable as soon as there is dilution. The distribution of the couplings shows the typical gap with a width depending on the amount of dilution, but this gap persists even in cases where a particular type of coupling plays no role in the learning process.