Studying dynamics of complex biological systems from structure of - PowerPoint PPT Presentation

Studying dynamics of complex biological systems from structure of networks Atsushi Mochizuki 1,2 1 Theor. Biol. Lab., RIKEN, 2 CREST JST, mochi@riken.jp

Research activity of Theoretical Biology Laboratory Somite segmentation Protein interaction KaiC dissociation P KaiA P promote KaiC KaiB KaiC KaiB KaiA bind bind P KaiA P KaiC KaiC Gene regulatory network unknown Gene Network Regulation Dynamics Plant leaf formation Pattern Formation 細胞分化の多様性 Cell migration Cytomembrane Retinal cone mosaic

Studying network systems based on theories to connect structure and dynamics A: Gene regulatory network B: Signal transduction Sato group, Sato lab. Hiroshima group, Sako lab. Structural theories ・ Linkage logic         for all  0 j J x t x t  j j t     implies     x x 0 t t  t ・ Structural sensitivity                   w        S j 0 w x          j m   x       m     CREST Program, JST       “Bio-Dynamics” C: Carbon metabolism Shirane group, Nakayama-lab.

Structural theories Linkage Logic Regulatory networks Network -> Key molecules Rate sensitivity analysis Chemical reaction networks Network -> Response of system In biology, information of network are very rich, but qualitative details of dynamics are very poor. Structural theories use information of networks only.

Structural theories Linkage Logic Network information Regulatory networks Experiment Network -> Key molecules ・ Network analysis ・ Measuring Rate sensitivity analysis ・ Perturbation Chemical reaction networks Prediction without Network -> Response of system assuming details In biology, information of network are very rich, but qualitative details of dynamics are very poor. Structural theories use information of networks only.

Structural Sensitivity We study response of a reaction system to a knockout/overexpression of enzyme. We found ... A A B B KO KO (1) Qualitative responses are determined from network, only. C C D D (2) Characteristic pattern of responses. E E F F J J - localization and hierarchy G G H H (3) A general law governing the pattern connecting network topology I I and system behavior. Mochizuki A. & Fiedler B. (2015) J. Theor. Biol. 367 , 189-202. Okada T. & Mochizuki A. Phys. Rev. Lett. (In Press)

Analyzing a biological system based on network structure only Theory Network structure -> genes for “cell-fate control” Experiment (1) Verify controllability (2) Obtain unexpected information of regulation => Total understanding for the system based on network Mochizuki, A., Fiedler, B. et al. J. Theor. Biol. (2013) 335 , 130-146 Imai, K., et al. Science (2006) 312 , 1183-1187.

gene regulatory network specifying cell fates epidermis brain nerve cord muscle notochord mesenchyme endoderm initial conditions cell‐cell interactions Imai, K., et al. Science (2006) 312 , 1183‐1187.

Hoya a sea squirt an ascidian a chordate animal

Cell differentiation and gene activities egg 16-cell stage 64-cell stage gastrula tailbud stage muscle mesenchyme epidermis brain Gene expression patterns reflect cell types.

Diversity of gene expressions from a regulatory network Regulatory network gene expression A state 1 B C time gene expression Dynamics of concentration (or gene activity) state 2        , u f u u d u A A A C A A        u f u d u B B A B B time        , u f u u d u C C A B C C

・truly functional as expected? ・possibly incomplete? epidermis brain nerve cord muscle notochord mesenchyme endoderm initial conditions cell‐cell interactions Imai, K., et al. Science (2006) 312 , 1183‐1187.

Studying dynamics from network structure alone Linkage: Generalization Information of argument set �� � � � � � � , � � � j i � � : Input set of k correspond to edge of network � ∈ � � k iff gene k activates itself. �� � � � � � � , � � � � � � � Decay condition � � � � � � , � � � � 0 use argument set only. do not consider detailed formula of f k , d k . (including self repression) Uniqueness

Collaboration with Mathematical concepts Bernold Fiedler (Berlin Free Univ.) Feedback vertex set � Determining nodes � in Graph theory in Dynamical theory          for all  0 j J V x t x t    j j 0 t is feedback vertex set I V implies if, and only if, \ is cycle free V I         x x 0 t t  t A set of vertices whose removal Observing long-term dynamics on leaves a graph without cycles determining nodes ⇒ Observing attractors of total system. A feedback vertex set � ⟺ a set of determining nodes � for any functions. Fiedler, B., Mochizuki, A. et al. J. Dyn. Differ. Eqns. (2013) 25 , 563-604. Mochizuki, A., Fiedler, B. et al. J. Theor. Biol. (2013) 335 , 130-146

The theory gives an assurance that: We detect all of the attractors by measuring “feedback vertex set” only. 1 node 2 nodes 1 node 1 node 1 node 3 nodes 1 node 2 nodes

Intuitive explanation 1. If dynamics of upward is given, dynamics of downward is determined uniquely. 2. Dynamics of appropriately selected sets determine dynamics of whole network. 3. How can we minimize the set, on which dynamics are given? ⇒ Feedback Vertex Set!

Control aspect of Feedback Vertex Set FVS ＝ Determining nodes J Observing long-term dynamics on Feedback Vertex Set ⇒ Observing attractors of total system.          for all  0 j J V x t x t  j j t implies Controlling Feedback Vertex Set to converge to one of attractors         x x 0 t t  t ⇒ Control of total system.

Controlling a dynamical system 2 stable oscillation （ P1 、 P2 ） - A system for mammalian circadian rhythms 1 unstable oscillation （ UP ） - 7 FVs among 21 variables 1 unstable stationary point （ USS ） Mirsky et al., 2009

Controlling a dynamical system Whole system can be controlled by controlling FVS. P1 -> P2 - 7 FVs among 21 variables FVS ＜＝ prescribe Others ＜＝ ODE (1) Prepare time track of FVS on the solution, P1, P2, UP, USS. (2-1) Prescribe FVS, to follow the value on the solution. (2-2) The remaining variables, nonFVS, are calculated by remaining ODEs.

Controlling a dynamical system Whole system can be controlled by controlling FVS. - 7 FVs among 21 variables FVS ＜＝ prescribe Others ＜＝ ODE UP, USS; unstable in original system with 21 variable, but stable in reduced system with 14 variables.

Controlling a dynamical system System can not be controlled by controlling subset of FVS. - 7 FVs among 21 variables FVS － CLK ＜＝ prescribe CLK ＜＝ ODE Others ＜＝ ODE

Dynamics of complex systems ⇔ Structure of regulatory network Uniqueness Feedback vertex set (determined from networks) (1) specify minimal sufficient set of variables to detect all of the possible dynamical behaviors. (2) specify minimal sufficient set of variables to control whole system. Fiedler, B., Mochizuki, A. et al. J. Dyn. Differ. Eqns. (2013) 25 , 563-604. Mochizuki, A., Fiedler, B. et al. J. Theor. Biol. (2013) 335 , 130-146

Structural Controlablity Liu et al. Nature (2011) �� � � �� � � �� � � � ��

Structural Control FVS Control �� � � �� � � �� � �� � � � � � � , � � � � � �� � � � � � 0 Linear Non-linear (require decay condition) Steering Switching between attractors ∗ � � � 0 � � � � � � Observing nodes � Controlling nodes Observing nodes = Controlling nodes (Observed data can be directly used for control) Diversity generated from internal Including effect of input signals dynamics only For details, See Mochizuki, A., Fiedler, B. et al. J. Theor. Biol. (2013) 335 , 130-146

gene regulatory network specifying cell fates epidermis brain nerve cord muscle notochord mesenchyme endoderm initial conditions cell‐cell interactions Imai, K., et al. Science (2006) 312 , 1183‐1187.

(1) Analysis for Ascidian network Remove genes regulating no gene (down most) genes not regulated by any genes (upper most) nodal Otx FoxD-a/b Twist-like-1 NoTrlc ZicL FGF9/15/20 1 feedback vertex Dynamics of 80 genes ⇒ Attractors are detected by only one gene If gene expressions are binary (on/off), the system can generate only two states. It is impossible to generate 7 different binary steady states (cell differentiations). “Binary” is incorrect, or Network is not enough.

gene regulatory network specifying cell fates Version 2015 epidermis brain nerve cord muscle notochord mesenchyme endoderm initial conditions cell‐cell interactions