Abstract
The central question of systems biology is to understand how individual components of a biological system such as genes or proteins cooperate in emerging phenotypes resulting in the evolution of diseases. As living cells are open systems in quasi-steady state type equilibrium in continuous exchange with their environment, computational techniques that have been successfully applied in statistical thermodynamics to describe phase transitions may provide new insights to the emerging behavior of biological systems. Here we systematically evaluate the translation of computational techniques from solid-state physics to network models that closely resemble biological networks and develop specific translational rules to tackle problems unique to living systems. We focus on logic models exhibiting only two states in each network node. Motivated by the apparent asymmetry between biological states where an entity exhibits boolean states i.e. is active or inactive, we present an adaptation of symmetric Ising model towards an asymmetric one fitting to living systems here referred to as the modified Ising model with gene-type spins. We analyze phase transitions by Monte Carlo simulations and propose a mean-field solution of a modified Ising model of a network type that closely resembles a real-world network, the Barabási–Albert model of scale-free networks. We show that asymmetric Ising models show similarities to symmetric Ising models with the external field and undergoes a discontinuous phase transition of the first-order and exhibits hysteresis. The simulation setup presented herein can be directly used for any biological network connectivity dataset and is also applicable for other networks that exhibit similar states of activity. The method proposed here is a general statistical method to deal with non-linear large scale models arising in the context of biological systems and is scalable to any network size.
Highlights
Biological networks are multi-dimensional complex systems whose collective interaction in response to perturbations may lead to critical transitions from one stable state to another
More examples of applications of Ising models to biological systems include, but is not limited to, a four-dimensional cellular automaton-like Ising model in which cells transition between normal, proliferative, hypoxic and necrotic states has been used to model the tumorigenesis process which involves a transition between these pre-malignant and malignant cell states (Durrett 2013; Torquato 2010); estimating information transfer between spins occurring in human connectome (Marinazzo et al 2014); the transition of B-DNA to S-DNA (Ahsan et al 1998); estimation of differentially expressed genes in cancer patients (Xumeng et al 2011); and approximation of join expression profiles of genes using a small number of observations (Santhanam et al 2009). To our knowledge, these models do not take into account two aspects of modeling phase transitions in biological networks that we address in this study
Collective flipping of coherently expressed genes is associated with disease progression
Summary
Biological networks are multi-dimensional complex systems whose collective interaction in response to perturbations may lead to critical transitions from one stable state to another. Most complex step scales as O(N 2) in the update schemes, and to our knowledge, the model is usually applied to networks whose sizes are of the order of hundred nodes (Campbell and Albert 2014; Wang et al 2012; Zhang et al 2008) This is because building such state transition graphs gets computationally expensive as network size increases due to the exponential dependence of the size of state space on network size, thereby making it challenging to analyze large-scale interconnected biological networks such as, for example, the complete human genome. The behavior of the system is characterized by the mean of the summed states of the system, referred to as magnetization in the context of the classical Ising model This method is an abstraction of the gene network and, only requires the initial configuration of all nodes of the network and network connectivity.
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