Journal of Theoretical Biology 302 (2012) 29–38 Contents lists available at SciVerse ScienceDirect Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/yjtbi Quantification of degeneracy in biological systems for characterization of functional interactions between modules Yao Li a, Gaurav Dwivedi b, Wen Huang c, Melissa L. Kemp b,n, Yingfei Yi a,d,1 a School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA b The Wallace H. Coulter Department of Biomedical Engineering, Georgia Institute of Technology and Emory University, Atlanta, GA 30332-0363, USA c University of Science and Technology of China, Hefei 230026, China d School of Mathematics, Jilin University, Changchun 130012, China article info abstract Article history: There is an evolutionary advantage in having multiple components with overlapping functionality (i.e Received 5 August 2011 degeneracy) in organisms. While theoretical considerations of degeneracy have been well established Received in revised form in neural networks using information theory, the same concepts have not been developed for 15 February 2012 differential systems, which form the basis of many biochemical reaction network descriptions in Accepted 20 February 2012 systems biology. Here we establish mathematical definitions of degeneracy, complexity and robustness Available online 28 February 2012 that allow for the quantification of these properties in a system. By exciting a dynamical system with Keywords: noise, the mutual information associated with a selected observable output and the interacting Degeneracy subspaces of input components can be used to define both complexity and degeneracy. The calculation Complexity of degeneracy in a biological network is a useful metric for evaluating features such as the sensitivity of Robustness a biological network to environmental evolutionary pressure. Using a two-receptor signal transduction Systems biology Biological networks network, we find that redundant components will not yield high degeneracy whereas compensatory mechanisms established by pathway crosstalk will. This form of analysis permits interrogation of large- scale differential systems for non-identical, functionally equivalent features that have evolved to maintain homeostasis during disruption of individual components. & 2012 Elsevier Ltd. All rights reserved. 1. Introduction ability of the modules to interact and perform functions coher- ently (functional integration). A highly complex system maintains In 1999, Hartwell et al. introduced the concept of modular segregation of function while still allowing for functional integra- biology, where a functional module is created by interacting tion. Degeneracy measures how well functionally independent molecules collectively performing a discrete function. Such mod- modules can interact to produce redundant outputs. Modules that ules provide an advantage for both evolvability (sensitivity to are structural duplicates form a completely redundant system environmental changes) and robustness (insensitivity to pertur- and always produce the same output; degenerate systems arise bations) in a system (Hartwell et al., 1999). Reconciling the from structurally distinct modules with different outputs inter- divergent requirements of evolvability and robustness requires acting to produce the same output under certain conditions. knowledge regarding the connections between these functional Systemic features like degeneracy, complexity, robustness are units and an appreciation for how each module integrates related to one another. It has already been observed via numerical information arriving from multiple inputs. In complex biological simulations for neural networks that high degeneracy not only systems such as neural networks, there has been recent emphasis yields high robustness, but also it is accompanied by an increase on features of structural complexity such as degeneracy. As first in structural complexity (Tononi et al., 1999). Thus, it is believed introduced in Tononi et al. (1994), structural complexity can be that degeneracy is a necessary feature in the evolution of complex understood in terms of the interplay between specialization of biological systems, partly because genetically dissimilar organ- functions in individual modules (functional segregation) and the isms will drive toward convergence of function while maintaining non-redundant components (Edelman and Gally, 2001). It is also believed that the robustness and adaptability that ensue from n Corresponding author. Tel.: þ1 404 385 6341. degeneracy are key features of complex biological systems at E-mail addresses: [email protected] (Y. Li), multiple scales (Stelling et al., 2004) and organisms have evolved [email protected] (G. Dwivedi), [email protected] (W. Huang), [email protected] (M.L. Kemp), [email protected] (Y. Yi). to contain many non-identical structures to produce similar 1 Senior author. functions. In this manner, degeneracy helps to fulfill the necessary 0022-5193/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2012.02.020 30 Y. Li et al. / Journal of Theoretical Biology 302 (2012) 29–38 properties of biologically functional modules. While the concept where Wt is the Wiener process, s is a non-singular, n  n matrix- of degeneracy was introduced for neural networks, there is strong valued function and E is a small parameter. The time evolution of evidence that many dynamical biological networks including the probability density function associated with the SDE (2) cellular metabolic and signaling networks exhibit the property satisfies the so-called Fokker–Planck equation of degeneracy; for example, protein kinase isoforms regulated by 1 Xn separate genes phosphorylate the same substrate protein r ¼ E2 ðA rÞ Àrðf rÞ, ð3Þ t 2 ij ij (Edelman and Gally, 2001). Furthermore, because regulatory i,j ¼ 1 features of protein or metabolic networks often rely on dynamical where AðxÞ¼sðxÞsT ðxÞ is an n  n symmetric non-negative defi- systems analysis, such theory must be compatible with kinetic nite matrix. description of biochemical reactions rather than neural network Of particular importance among the solutions of the Fokker– (Tononi et al., 1999) or logic-based description (Rizk et al., 2009). Planck equation are the steady states, which satisfy the stationary Although some features like regulation and robustness of bio- Fokker–Planck equation chemical networks of signal transduction have been studied 8 Xn quantitatively (Kitano, 2007; Rizk et al., 2009), the features of > 1 < E2 ðA rÞ Àrðf r Þ¼0, interest here, such as degeneracy and complexity, have not been 2 ij ij E > i,j ¼ 1 ð4Þ formalized mathematically in terms of ordinary differential equa- :> R rðxÞ40, n r ðxÞ dx ¼ 1: tion description that is so commonly used for describing protein R E networks. By developing a method for calculating degeneracy in A smooth solution rE of the stationary Fokker–Planck equation (4) general biological networks modeled by differential equations, is known to uniquely exist (Bogachev et al., 2009; Huang et al., network structures can be explored that fulfill the contrary 2011)if requirements of evolvability and robustness first posited by Hartwell et al. Understanding these features becomes important f is differentiable and s is twice differentiable on Rn; and as increasingly complex biological systems are being designed ab ð Þ -þ1 thereP exists a Lyapunov function V x 40, V such that initio in synthetic biology. In this article, we first mathematically 1 2 n 2 E AijðxÞ@ VðxÞþf ðxÞVðxÞoÀg for some constant g40 formalize the relationships between degeneracy, complexity and 2 i,j ¼ 1 ij 9 9 robustness in dynamical systems. We next establish that high and all x sufficiently large. degeneracy always yields high complexity. Finally, we illustrate two distinct approaches for numerically calculating degeneracy in We remark that while the ODE (1) may have many complicated dynamical systems, a predator–prey model and a kinetic signaling invariant measures without even having density functions, the model. In many instances, the metric of degeneracy is a useful steady-state of the SDE (2) is nevertheless unique and smooth. indicator of how easily the system adapts under evolutionary We also note that when E is fixed, we denote the invariant pressure. solution as r instead of rE. 2.2. Definitions of degeneracy, complexity and robustness 2. Mathematical approach 2.2.1. Degeneracy and complexity 2.1. Random perturbations of ODE system Inspired by, but differing from Tononi et al. (1999), our definition of degeneracy is divided into several steps. In the first Our strategy is to inject a fixed amount of stochastic perturba- step, we define projected density, entropy and mutual informa- tion into a differential system. With such small random perturba- tion associated with any subspace. Then, we fix a subspace as the tions, the corresponding variable sets of modules of the network ‘‘output’’ set and define its associated degeneracy by considering become stochastic processes. If two modules have strong func- the complementary subspace as the ‘‘input’’ set. Lastly, we define tional connectivity, then these two stochastic processes should the degeneracy of the entire system by varying the output sets have high statistical correlation. Conversely, two functionally and taking the maximum among all degeneracies of these sets. independent components must be statistically independent. This Definitions of projected density, entropy