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Flexible and robust networks: S. A. Vakulenko (S. Petersburg, Institute for Mechanical Engineering Problems, Russian Academy Sci., St. Petersburg State University of Technology and Design ) O Radulescu (University Montpellier II, Dept. of Biology) Some results conjoint with D. Grigoriev (Lille), M.Zimin (St Petersburg) M. Cherkay (St Petersburg) S. Genieys (Tolouse) V. Gursky (S Petersburg).

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Flexibility and robustness Flexibility and robustness are important properties of living systems in general, most particularly observed during development from egg or embryo to a fully organized organism. Flexibility means the capacity to change, when environmental conditions change. Opposite to this, robustness is a capacity to support homeostasis in spite of external changes.

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GOAL Is to confirm mathematically the following Biological idea: microRNA maintain pattern robustness with respect to perturbations We consider network architexture which is effective in pattern robustness

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MicroRNA and robustness A MicroRNA Imparts Robustness against Environmental Fluctuation during Development Xin Li,1,2,3 Justin J. Cassidy,1,2 Catherine A. Reinke,1 Stephen Fischboeck,1 and Richard W. Carthew1,* 1Department of Biochemistry, Molecular Biology and Cell Biology, 2205 Tech Drive, Northwestern University, Evanston, Illinois 60208, USA

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In more details The conceptual significance of the robustness-miRNA connection is several-fold. Their dynamic kinetic properties help answer the question of ‘‘why miRNA gene regulation’’ instead of just using more transcription factors. Their rate of biogenesis is more rapid than proteins, and they affect expression with less delay than factors that regulate nuclear events. These features enable miRNAs to produce rapid responses, something that is expected to counteract rapid and variable fluctuations. It also explains why miRNAs frequently appear dispensable under uniform laboratory conditions (Bushati and Cohen, 2007; Leaman et al., 2005; Miska et al., 2007).

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Canalization microRNAs as Canalization Factors Waddington coined the word canalization to describe how development is buffered against perturbation (Siegal and Bergman, 2002; Waddington, 1942). Despite considerable genetic or environmental variation, organisms develop traits that are remarkably uniform in phenotype. Indeed, the insect compound eye and sensory organs appear to be deeply canalized systems (Jander and Jander, 2002; Meir et al., 2002; Rendel, 1959). It has been speculated that miRNAs might be important for canalization (Hornstein and Shomron, 2006). Certainly miR-7 has many attributes that suggest it helps canalize development in Drosophila.

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Mathematical tools are developed in series of Publications Vakulenko S. and Genieys S., ( 2005) Patterning by genetic networks, Mathematical Methods in Applied Sciences, 29, Vakulenko S., Grigoriev D., ( 2006) Algorithms and complexity in biological pattern formation problems, Annales of Pure and Applied Logic 141, Vakulenko S., Grigoriev D., (2005) Evolution in random environment and structural stability, Zapiski seminarov POMI RAN,325, Review: Vakulenko S., Grigoriev D., Еvolution, Instability and Morphogenesis, Chapter in a book, Nova Publishers, New York 2008.

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Publications Vakulenko S. A., (2000) Dissipative systems generating any structurally stable chaos,Advances in Diff. Equations, 5, Vakoulenko, S. Complexite dynamique de Reseaux de Hopfield, (2002) C. R. Acad. Sci. Paris Ser. I Math., t.335. O. Radulescu, Vakulenko S.,Manu, J. Reinitz, Phys. Rev. Letters, 2009 RECENT PAPERS : Vakulenko S. and Zimin M., (2010) Int. Journal of Nanotechnology and Molecular Computation, С. Вакуленко М.Черкай Теор и Матем Физика (2010) Vakulenko S., M. Zimin., (2010) submitted in J Physic A Matem. Theoret ical

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Reinitz Sharp Mjolness model ∂u i (x, t) /∂t = d i Δu i (x, t) + r i σ (Σ k K ik u k - h i ) - λ i u i, k,i=1,2,…,N N- number of nodes σ –so called sigmoidal function, for example, σ =1/(1 + exp(-ax)) Here we take into account three main effects: Pair interaction of nodes (activation or inhibition), Degradation (dissipation) (term with λ ) Diffusion For d i =0 this reduces to the Hopfield model, 1982, Figotin-Pastur The Hopfield is a basic model of attractor neural network theory

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Sigmoid function The logistic curvelogistic curve

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Simplified time discrete version (2) u i (t+1) = σ (Σ k K ik u k (t) - h i ) j, k=1,2,…,N Matrix K defines an interaction between nodes

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Graphs We can associate a graph (V,E) with the interaction matrix K ik V={1,2,…,N} Edge (i, k ) E if and only if K ik ≠ 0 Assume (V,E) has so-called free scale structure : Prob ( a vertex i has k adjacent vertices) =const k -γ 2 < γ < 3

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Attractor complexity If K is symmetric, the network attractor is a union of rest points (Hopfield 1982) If K is an arbitrary, these networks can generate any structurally stable attractors (Vakulenko ) depending on N,K, h. They can simulate any Turing machines (P. Koiran et al, 1999 )

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Main idea Synergetics: Slaving principle Dynamics of fast modes is captured by dynamics of slow modes In networks slow modes are states of hubs Fast modes are states of satellites of these hubs

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Scale free structure supports multistationarity and complexity Large networks with scale free structure can generate complicated attractors (possibly, periodic or chaotic) The networks can generate a number of local attractors Multistationarity can help in adaptation

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Simple model of a flexible robust network As a math. model, we use the following equations du i (t) /dt = r σ ( bq - h i ) - λ u i, dq (t) /dt = ε(σ (Σ k w k u k - h) - aq), Green variables u i are fast, blue variable q is slow, h are morphogenetic fields We interprete mobile u i as microRNA, center q as a TF

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Network topological structure

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Method If r, ε are small, λ >0 is not small Then we can express u i via q. After we substitute this expression into v-equation And seek stationary solutions defined by aq =f(q). Solutions are intersections of the right line and the graph of f(q)

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q-dynamics captures u-dynamics: We can express u via q. Blue curve is f(q), red one: aq

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Another, simpler variant: Figure 1. The green curve is the plot of f(q). The red line is the plot of aq + h, intersections give equilibria. q lies on the horizontal axe

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Even strong perturbations in thresholds h i conserve old equilibria, but we can obtain new ones

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Bifurcations Bifurcations of network dynamics can be investigated easily for n=1( single center) Suppose we eliminate a weakly connected node or two such nodes Then we observe different changes of q- map. Next images show perturbations in location of rest points (obtained by Matlab)

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RNA elimination does not change dynamics

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RNA elimination sharply change dynamics

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N of miRNA and K of TF Numerical experiments (M. Cherkay) and Some analytical ideas show that the number of stationary states Has order N^K.

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Complexity of bifurcations of network dynamics There are possible many different situations A) elimination of many nodes can weakly change dynamics B) elimination of a special node can drastically change dynamics

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Conclusions Genetic networks with natural and simple structure can exhibit complicated dynamics, in particular, multistationarity; 1)To be robust, they should consist of many small mobile elements and large slow elements; 2) To increase robustness, networks evolve increasing complexity of interaction graph; 3) However, too complicated networks become too slow (relaxation time is large)

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Last conclusion Last conclusion can be understood by an example of a great Empire with a single center and many independent provinces (regions). Regions do not interact (divide et impera!). Then the center can control all regions if the center mobility is less than speeds of region evolution. We obtain an alternative: either empire lost control, or empire evolves too slow.

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