INDIANA UNIVERSITY …...dynamical models of regulation from qualitative data Albert & Othmer...

[email protected]://informatics.indiana.edu/rocha

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biological interpretations of attractor behavior

Genetic regulatory networks Genes are on or off Development, morphogenesis Attractors interpreted as different cell types

Classification in Immune networks Representation in artificial neural networks Stable patterns of species abundances in ecosystems

self-organization

attractors spontaneous order To be improved by natural selection

Order is the raw material for evolution: how much of life is Natural Selection and how much is self-organization? (credit assignment problem)


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Self-organization

Emergent Behavior from system/environment coupling Classifies Walls and Other

Robots Self-organization Embodied cognition

Robot example

http://www.johuco.com/muram/muram.html

Jonathan Connell ‘s Muramator


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canonical complex systemsredundancy in networks with dynamics

NK Boolean Network (N=13, K=3) Multivariate Dynamical Systems:Structure: Variable interactions, associations, influenceDynamics: variable states (micro) network configurations (macro)Redundancy: links (path backbones), state transitions (canalization)

Minimal networks with both structure and dynamics.

Interactions and variables with binary states. Huge state-

spaces and ensembles for same structure. Full range of

attractor behavior

possible Boolean functions of k inputs (k=3 →256)K22 Network configurations (state-space)N2

Kauffman, SA. J. theoretical biology 22.3 (1969): 437-467.

N

i

ik

1

22# different Boolean networks for same structure (25613)


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discrete dynamical systemsExample 13 variables

NK Boolean Network (N=13, K=3)DDLab (Andy wuensche): http://www.ddlab.com/

There are 213=8192 possible states but only a small set of attractors


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[email protected]/rocha

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the drosophila segment polarity networkdynamical models of regulation from qualitative data

Albert & Othmer [2003]. J. Theor. Bio. 223: 1-18.

Based on the ODE model of von Dassow et al. (2000), consists of 4-cell parasegments, each cell with 15 interacting genes and proteins.

260 network configurationsReproduces wild-type and mutant gene expression patterns in development of fruit fly

2 intercellular inputs: nhh (hedgehog), nWG (wingless)1 intracellular input: SLP (sloppy paired)Reka Albert

Barman & Kwon.PloS one 12.2 (2017): e0171097.


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Kauffman’s statistical analysis

Random networks Started with random initial conditions

Self-organization is not a result of special initial conditions

Statistical analysis K 2

Steady state, ordered, crystallization (5 K to ) K=N

Disordered, chaotic Mean length of cycles: 0.5 x 2N/2

Mean number of cycles: N/e High reachability, sensitive to perturbation

Number of other state cycles system can reach after perturbation

K=2 Mean length: n1/2

Mean number of cycles: n1/2

Low reachability Percolation of frozen clusters (isolated subsets) Not very sensitive to perturbation

Of NK-Boolean Networks


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edge of chaos on Boolean Networks

2 K 5 Good for evolvability? Some changes with large repercussions Best capability to perform information exchange

Information can be propagated more easily Problems with analysis

Network topology is random Not scale-free, as later explored by Aldana

Real genetic networks tend to have lower values of K (in ordered regime)

Genes as simply Boolean may be oversimplification Though a few states can approximate very well continuous

data

criticality


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dynamical behavior of ensembles of networkscriticality in Boolean networks

Aldana, M. [2003]. Physica D. 185: 45–66

gyRandom topology

gyscale-free topology𝑃 𝑘 𝑐.𝑘


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current theory (homogenous networks)criticality in Boolean networks

Manicka, Marques-Pita, & Rocha, [2020]. In Prep.

kp 211

21


Manicka, S., [2017]. PhD. Dissertation, Indiana University..

Derrida & Pomeau. [1986] EPL . 1.2: 45.

2.𝑘.𝑝 1 𝑝 12.𝑘.𝑝 1 𝑝 1


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Cellular automatahomogenous lattice of state-determined systems

xx-1 x+1

Cellular Automata

xt

2-D

},...2,1,0{

,...,..., 1

sxxxxfx t

riiriti

1-D

x

1,,,, ,...,..., t

rjrijirjrit

ji yxyxxxxfx

Toroidal LatticeToroidal LatticeToroidal Lattice

Space-time diagram


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cellular automata

Parallel updating Artificial physics

Local interactions only No actions at a distance

Homogeneous Unpredictable global behavior

Emergence 2-levels: rules (micro-level) and

attractor behavior (macro-level) Irreversible

Self-organization Example rules

Rug (diffusion) 256 states Average of 8 neighbors in 2-d grid, if

state is 255 -> 0. Vote/majority

binary


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elementary CA rules Radius 1

Neighborhood =3 Binary

23 = 8 input neighborhoods 28 = 256 rules

http://mathworld.wolfram.com/CellularAutomaton.html

xx-1 x+1

Cellular Automata

xt


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state-determined transitionsCellular automata


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Living patterns easily replicated in CA


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What’s a CA?more formally

D-dimensional lattice L with a finite automaton in each lattice site (cell)

State-determined system

finite number of states Σ: K=| Σ|E.g. Σ = {0,1}

finite input alphabet α

transition function Δ: α→Σ

uniquely ascribes state s in Σ to input patterns α

Neighborhood templateN

NN K ,

Number of possible neighborhood states

NKKD

Number of possible transition functions

Example

K=8

N=5

|α|=37,768

D 1030,000

Example (ECA)

K=2, N=3,

|α|=23=8

D = 28=256


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Langton’s parameter

Statistical analysis Identify classes of transition functions with similar behavior

Similar dynamics (statistically) Via Higher level statistical observables

Like Kauffman The Lambda Parameter (similar to bias in BN)

Select a subset of D characterized by λ Arbitrary quiescent state: sq

Usually 0 A particular function Δ has n transitions to this state and (KN-n)

transitions to other states s of Σ (1-λ) is the probability of having a sq in every position of the rule table

Finding the structure of all possible transition functions

Langton, C.G. [1990]. “Computation at the edge of chaos: phase transitions and emergent computation”. Artificial Life II. Addison-Wesley.

N

N

KnK

λ = 0: all transitions lead to sq (n =KN)

λ = 1: no transitions lead to sq (n =0)

λ = 1-1/K: equally probable states ( n=1/K . KN)Range: from most homogeneous to most

heterogeneous


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Langton’s observations λ only correlates well with dynamic behavior for fairly large values of K and N

E.g. K≥4 and N≥5 Experiments

K=4, N=5


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Langton’s results


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Langton’s results

Approximate time when density is within 1% of long-term behavior


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Langton’s results



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Langton’s results



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Edge of chaos

Transient growth in the vicinity of phase transitions Length of CA lattice only relevant around phase transition (λ=0.5)

Conclusion: more complicated behavior found in the phase transition between order and chaos Patterns that move across the lattice

A phase transition?


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Computation at the edge of chaos?

Transition region Supports both static and propagating structures

λ =0.4+ Propagating waves (“signals”?) across the CA lattice

Necessary for computation? Signals and storage?

Computation Requires storage and transmission of information Any dynamical system supporting computation must exhibit

long-range signals in space and time Wolfram’s CA classes

I: homogeneous state Steady-state

II: periodic state Limit cycles

III: chaotic IV: complex patterns of localized structures

Long transients Capable of universal computation


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Bias and lambda parameterEdge of chaos

xx-1 x+1

Cellular Automata


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Bias and lambda parameterEdge of chaos

xx-1 x+1

Cellular Automata


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from evolutionary robustness to network and dynamical redundancycanalization as a key mechanism for resilience

Waddington CH (1942). Nature. 150 (3811): 563–565

Kauffman, S. A. (1984). Phys. D Nonlinear Phen. 10, 145–156.

dynamics of gene networks provides buffering (self-organization). But still easily chaotic.

robustness of phenotypes is the result of a buffering of the developmental process.

genetic control ignores some inputs (redundancy) to attain necessary resilience (tradeoff stability/evolvability)


Structure (topological organization), can provide larger stable or critical universe, but still easily chaotic.


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from evolutionary robustness to network and dynamical redundancycanalization as a key mechanism for resilience

Waddington CH (1942). Nature. 150 (3811): 563–565

Kauffman, S. A. (1984). Phys. D Nonlinear Phen. 10, 145–156.

dynamics of gene networks provides buffering (self-organization). But still easily chaotic.

robustness of phenotypes is the result of a buffering of the developmental process.

genetic control ignores some inputs (redundancy) to attain necessary resilience (tradeoff stability/evolvability)


Structure (topological organization), can provide larger stable or critical universe, but still easily chaotic.


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k

Ff f

s

fxk

2

max)(

"

|

kFf f

r

fxk

2

#max "

|

quantifying redundancy (canalization) in automata nodes

Measuring two forms of canalization Kr= 2 Ke= 6 - 2 = 4 Ks = 4

input redundancy/effective connectivity and input symmetry

Marques-Pita & Rocha, [2013]. PLoS ONE, 8(3): e55946.

kr(x) = mean number of “#” in LUT

xkxkxk re )()(

6)( xk

Prime Implicants (Quine-McCluskey)

minimal transition control: set of wildcard schemata is DNF of prime implicants (Blake Canonical Form)

p: bias, ratio of “1’s” in output

p(x) = 14/64 ≈ 0.22

plus group invariance ks(x) = mean number of “○” in LUT


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k

Ff f

s

fxk

2

max)(

"

|

kFf f

r

fxk

2

#max "

|

quantifying redundancy (canalization) in automata nodes

Measuring two forms of canalization Kr= 2 Ke= 6 - 2 = 4 Ks = 4

input redundancy/effective connectivity and input symmetry


kr(x) = mean number of “#” in LUT

xkxkxk re )()(

6)( xk

Prime Implicants (Quine-McCluskey)

minimal transition control: set of wildcard schemata is DNF of prime implicants (Blake Canonical Form)

p: bias, ratio of “1’s” in output

p(x) = 14/64 ≈ 0.22

plus group invariance ks(x) = mean number of “○” in LUT

f(xi)


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effective connectivity in evolution and bioengineering for stabiltycriticality in the presence of canalization/redundancy

xkxkxk re )()(


c.𝑘 .𝑝 1 𝑝 1c.𝑘 .𝑝 1 𝑝 1c.𝑘.𝑝 1 𝑝 1c.𝑘.𝑝 1 𝑝 1

Manicka, Marques-Pita, & Rocha, [2020]. In Prep.Manicka, S., [2017]. PhD. Dissertation, Indiana University..

Aldana, M. [2003]. Physica D. 185: 45–66Derrida & Pomeau. [1986] EPL . 1.2: 45.


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effective connectivity in evolution and bioengineering for stabiltycriticality in the presence of canalization/redundancy

xkxkxk re )()(


c.𝑘 .𝑝 1 𝑝 1c.𝑘 .𝑝 1 𝑝 1c.𝑘.𝑝 1 𝑝 1c.𝑘.𝑝 1 𝑝 1

Manicka, Marques-Pita, & Rocha, [2020]. In Prep.Manicka, S., [2017]. PhD. Dissertation, Indiana University..

Stable biochemical networks can exist well into expected chaotic behavior, provided canalization is selected for: dynamics

“buffers” underlying interaction structure New theory uses only dynamical

redundancy properties

Aldana, M. [2003]. Physica D. 185: 45–66Derrida & Pomeau. [1986] EPL . 1.2: 45.


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low effective connectivity leads networks closer to “edge of chaos”ubiquitous canalization in systems biology models

Manicka, Marques-Pita, & Rocha, [2020]. In Prep.Manicka, S., [2017]. PhD. Dissertation, Indiana University.

63 Biochemical regulation models with very low effective connectivity despite high

connectivity. In new theory networks are near criticality


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INDIANAUNIVERSITY

low effective connectivity leads networks closer to “edge of chaos”ubiquitous canalization in systems biology models

Manicka, Marques-Pita, & Rocha, [2020]. In Prep.Manicka, S., [2017]. PhD. Dissertation, Indiana University.

63 Biochemical regulation models with very low effective connectivity despite high

connectivity. In new theory networks are near criticality

c.𝑘.𝑝 1 𝑝 1c.𝑘.𝑝 1 𝑝 1 c.𝑘 .𝑝 1 𝑝 1c.𝑘 .𝑝 1 𝑝 1

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