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

Studying dynamics of complex biological systems from structure of networks Atsushi Mochizuki1,2

• 1Theor. Biol. Lab., RIKEN,

2CREST JST,

mochi@riken.jp

Gene Regulation Pattern Formation

Research activity of Theoretical Biology Laboratory

KaiC KaiA KaiB P KaiC KaiA P KaiC KaiA KaiB P KaiC promote bind bind dissociation P KaiC unknown

Gene regulatory network 細胞分化の多様性 Protein interaction Plant leaf formation Retinal cone mosaic

Network Dynamics

Somite segmentation Cell migration Cytomembrane

Studying network systems based on theories to connect structure and dynamics

Structural theories

・Linkage logic ・Structural sensitivity

A: Gene regulatory network

Sato group, Sato lab.

   

j j t

x t x t



    j J 

for all implies

   

t

t t



   x x 

B: Signal transduction

Hiroshima group, Sako lab.

C: Carbon metabolism

Shirane group, Nakayama-lab.

w w x x                                                     S

CREST Program, JST “Bio-Dynamics”

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

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. Experiment

・Network analysis ・Measuring ・Perturbation Network information Prediction without assuming details

A B C D E F G H J I

KO

A B C D E F G H J I

KO

Structural Sensitivity

We found ... (1) Qualitative responses are determined from network, only. (2) Characteristic pattern of responses.

• localization and hierarchy

(3) A general law governing the pattern

connecting network topology and system behavior.

We study response of a reaction system to a knockout/overexpression of enzyme.

Mochizuki A. & Fiedler B. (2015) J. Theor. Biol. 367, 189-202. Okada T. & Mochizuki A. Phys. Rev. Lett. (In Press)

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

Analyzing a biological system based on network structure only

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

brain notochord epidermis muscle mesenchyme endoderm nerve cord 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

muscle mesenchyme epidermis brain

16-cell stage 64-cell stage gastrula egg tailbud stage

Gene expression patterns reflect cell types.

           

, ,

A A A C A A B B A B B C C A B C C

u f u u d u u f u d u u f u u d u         

Diversity of gene expressions from a regulatory network

B C A Regulatory network Dynamics of concentration (or gene activity)

time gene expression gene expression time

state 1 state 2

brain notochord epidermis muscle mesenchyme endoderm nerve cord initial conditions cell‐cell interactions

・truly functional as expected? ・possibly incomplete?

Imai, K., et al. Science (2006) 312, 1183‐1187.

Linkage: Information of argument set

use argument set only. do not consider detailed formula of fk, dk.

Uniqueness

Studying dynamics from network structure alone

i j k

, : Input set of k correspond to edge of network iff gene k activates itself. Decay condition

Generalization

(including self repression) ,

• ∈
• , 0

Mathematical concepts

   

j j t

x t x t



   

Collaboration with Bernold Fiedler (Berlin Free Univ.) I V   \ V I is feedback vertex set is cycle free if, and only if, j J V  

for all

implies

   

t

t t



   x x 

A feedback vertex set ⟺ a set of determining nodes for any functions.

A set of vertices whose removal leaves a graph without cycles

Feedback vertex set

in Graph theory

Determining nodes

in Dynamical theory

Observing long-term dynamics on determining nodes ⇒ Observing attractors of total 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

1 node 1 node 1 node 2 nodes 2 nodes 1 node 1 node The theory gives an assurance that:

We detect all of the attractors by measuring “feedback vertex set” only.

3 nodes

• 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?

Intuitive explanation

⇒ Feedback Vertex Set!

Control aspect of Feedback Vertex Set

   

j j t

x t x t



   

j J V  

for all

implies

   

t

t t



   x x 

Observing long-term dynamics

• n Feedback Vertex Set

⇒ Observing attractors of total system.

FVS ＝ Determining nodes J

Controlling Feedback Vertex Set to converge to one of attractors ⇒ Control of total system.

2 stable oscillation （P1、P2） 1 unstable oscillation （UP） 1 unstable stationary point （USS）

Controlling a dynamical system

• A system for mammalian circadian rhythms
• 7 FVs among 21 variables

Mirsky et al., 2009

FVS ＜＝ prescribe Others ＜＝ ODE

Controlling a dynamical system

• 7 FVs among 21 variables

P1 -> P2

(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.

Whole system can be controlled by controlling FVS.

FVS ＜＝ prescribe Others ＜＝ ODE

Controlling a dynamical system

Whole system can be controlled by controlling FVS.

• 7 FVs among 21 variables

UP, USS; unstable in original system with 21 variable, but stable in reduced system with 14 variables.

FVS － CLK ＜＝ prescribe CLK ＜＝ ODE Others ＜＝ ODE

System can not be controlled by controlling subset of FVS.

Controlling a dynamical system

• 7 FVs among 21 variables

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.

Dynamics of complex systems ⇔ Structure of regulatory network

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

Linear Steering Observing nodes Controlling nodes Including effect of input signals For details, See Mochizuki, A., Fiedler, B. et al. J. Theor. Biol. (2013) 335, 130-146

FVS Control

,

• Non-linear (require decay condition)

Switching between attractors

• ∗

Observing nodes = Controlling nodes

(Observed data can be directly used for control)

Diversity generated from internal dynamics only

gene regulatory network specifying cell fates

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

Dynamics of 80 genes ⇒ Attractors are detected by only one gene

(1) Analysis for Ascidian network

1 feedback vertex

nodal FoxD-a/b Otx Twist-like-1 ZicL FGF9/15/20 NoTrlc

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). Remove genes regulating no gene (down most) genes not regulated by any genes (upper most) “Binary” is incorrect, or Network is not enough.

gene regulatory network specifying cell fates

brain notochord epidermis muscle mesenchyme endoderm nerve cord initial conditions cell‐cell interactions Version 2015

A cell B cell

FVS genes

Thm： Whole system can be controlled by controlling FVS genes only. ・Verify whether cell fate can be controlled by changing expression of the five genes. 4 genes from each cell 1 gene from one of neighboring cells

Zic Foxd Twist-like1 Foxa Nodal Snail Neurogenin Delta-like dpERK1/2

5 species of genes

Theory 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.

Dynamics of complex systems ⇔ Structure of regulatory network

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

Experiment Gene network for Cell-fate determination (1) Verify controllability by FVS genes (2) Obtain regulatory mechanism for exclusive cell-fate

Kazuki Maeda (RIKEN -> Konan Univ.) Bernold Fiedler (Free Univ. Berlin) Yutaka Sato (Kyoto Univ.) Kenji Kobayashi (Kyoto Univ.)

Collaborators