1

Signal Transduction, Cellerator, and The Computable Plant

Bruce E Shapiro, PhD

bshapiro@caltech.eduhttp://www.bruce-shapiro.com/cssb

2

Overview

• Cellerator• Chemical Kinetics• Signal Transduction Networks

– Modular outlook– Switches, oscillators, cascades, amplifiers, etc.– Deterministic vs. Stochastic simulations

• Multicellular systems– Synchrony, pattern formation– The Computable Plant project

• Model Inference

3

Cellerator

4

Short Cellerator Demonstration

5

• Canonical form of a chemical reaction:

• Ri,Pi: Reactants, Products• sPi,sRi: Stoichiometry• k: Rate Constant• Example:• Law of Mass Action: The rate of the reaction is proportional to

the product of the concentrations of the reactants.

•

Law of Mass Action

€

sRixii=1nR∑ k ⏐ → ⏐ sPiyii=1

nP∑

€

BrO3+ HBrO2k 3 ⏐ → ⏐ 2HBrO2+ 2Ce

6

Law of Mass Action (2)

• Formal statement (for a single reaction):

• Interpretation of

€

BrO3+ HBrO2k 3 ⏐ → ⏐ 2HBrO2+ 2Ce

€

d

dt[BrO3] = (0 −1)k3[BrO3][HBrO2] = −k3[BrO3][HBrO2]

d

dt[HBrO2] = (2 −1)k3[BrO3][HBrO2] = k3[BrO3][HBrO2]

d

dt[Ce] = (2 − 0)k3[BrO3][HBrO2] = 2k3[BrO3][HBrO2]

€

d[X]

dt= (sPX

− sRX) [Ri ]

sRi

i=1

nR

∏

7

€

BrO3+ Br k1 ⏐ → ⏐ HBrO2+ HOBr

HBrO2+ Br k 2 ⏐ → ⏐ 2HOBr

BrO3+ HBrO2k 3 ⏐ → ⏐ 2HBrO2+ 2Ce

2HBrO2k 4 ⏐ → ⏐ BrO3+ HOBr

Ce k 5 ⏐ → ⏐ Br

⎧

⎨

⎪ ⎪ ⎪

⎩

⎪ ⎪ ⎪

⎫

⎬

⎪ ⎪ ⎪

⎭

⎪ ⎪ ⎪

⇒d[HBrO2]

dt= k1[BrO3][Br] - k2[HBrO2][Br]

+ k3[BrO3][HBrO2] - k4[HBrO2]2

Law of Mass Action (3)

• Add rates for multiple reactions

“Oregonator”

8

Cellerator Input for Oregonator

stn={{BrO3+Br HBrO2+HOBr, k1}, {HBrO2+Br 2*HOBr, k2},

{BrO3+HBrO2 HBrO2+2*Ce, k3},{2*HBrO2 BrO3+HOBr, k4},

{Ce Br, k5}};interpret[stn, frozen {BrO3}];

Hold BrO3 concentration Fixed

Stoichiometry

Rate Constants

9

Cellerator Output for Oregonator

{

{Br’[t]==-k1*Br[t]*BrO3[t]+k5*Ce[t]-

k2*Br[t]*HBrO2[t], Ce’[t]==-k5*Ce[t]+2*k3*BrO3[t]*HBrO2[t], HBrO2’[t]==k1*Br[t]*BrO3[t]-

k2*Br[t]*HBrO2[t]+k3*BrO3[t]*HBrO2[t] -k4*HBrO2[t]^2,

HOBr’[t]==2*k2*Br[t]*HBrO2[t]+k4*HBrO2[t]2+k1*Br[t]*BrO3[t]},

{Br, Ce, HBrO2, HOBr}}

List of Differential Equations and Variables

10

Cellerator Simulation

s= predictTimeCourse[stn,frozen {BrO3},timeSpan 1500, rates {k1 1.3, k2 2*106, k3 34,

k4 3000., k5 0.02, BrO3[t] .1},

initialConditions {HBrO2 .001,Br .003,Ce .05,BrO3 .1};

{{0, 1500, {{Br InterpolatingFunction[{{0., 1500.}}, <>], Ce InterpolatingFunction[{{0., 1500.}}, <>], HBrO2 InterpolatingFunction[{{0., 1500.}}, <>], HOBr InterpolatingFunction[{{0., 1500.}}, <>]}}}}

Inpu

t

Out

put

11

Plot Results of Simulation

runPlot[s, plotVariables {Br},PlotRange { {400, 1400}, {0, 0.002}},TextStyle {FontFamily -> Times, FontSize -> 24}, PlotLabel "Br Concentration"];

OptionalInput

12

Basic Syntax

€

network = {{reaction, rate constants},

{reaction, rate constants},...}

reaction = reactants arrow productsmodifiers

modifiers

arrow = → or ⇒ or a or E

• Format of rate constants varies for different arrows• Modifiers are optional• Different rate laws for different arrow/modifier combinations• We will focus on reaction• Generate differential equation by entering

interpret[network]

13

Basic Mass Action Reactions

€

S → P

S1 + S2 +L → P1 + P2 +L

m1S1 + m2S2 +L → n1P1 + n2P2 +L

S +L E P +L means S +L → P +L and P +L → S +L

Cellerator Syntax:

€

{S → P, k}

{S1 + S2 +L → P1 + P2 +L ,k}

{m1S1 + m2S2 +L → n1P1 + n2P2 +L , k}

{S +L E P +L ,k1,k2}

We will generally omit explicitly writing the rate constants in the remainder of this presentation.

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Catalytic Mass Action Reactions

€

S + C → P + C becomes

€

S→C

P

€

S + CE SC → P + C or

S +C → SC

SC →S +C

SC → P +C

⎧

⎨ ⎪

⎩ ⎪

⎫

⎬ ⎪

⎭ ⎪becomes

€

SEC

P

€

S + CE SC → P + C

P + RE PR → S + R

⎧ ⎨ ⎩

⎫ ⎬ ⎭ or

S +C → SC

SC →S +C

SC → P +C

P +R → PR

PR → P +R

PR →S +R

⎧

⎨

⎪ ⎪ ⎪

⎩

⎪ ⎪ ⎪

⎫

⎬

⎪ ⎪ ⎪

⎭

⎪ ⎪ ⎪

becomes

€

SER

CP

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Cascades

€

A1 → A2 → A3 →L means A1 → A2, A2 → A3,L

A1E A2E A3E L means A1E A2, A2E A3,L

A1 → A2 → A3 →LC

means A1 → A2

C, A2 → A3

C,K

A1 → A2 → A3 →LC1 ,C 2 ,...

means A1 → A2

C1

, A2 → A3

C 2

,K

A1E A2E A3 →LC

means A1E A2

C, A2E A3

C,K

A1E A2E A3 →LC1 ,C 2 ,...

means A1E A2

C1

, A2E A3

C 2

,K

A1E A2E A3 →LR

F means A1E A2

R

F, A2E A3 →L

R

F

16

Michaelis-Menten Kinetics

• Catalytic Reaction:

• Mass action

• Steady-state assumption:

where E0 is total catalyst (bound + unbound)

€

SEC

P (i.e., S + C Ed

aSC→

kP + C)

€

d[S]

dt= −a[S][C] + d[SC],

d[P]

dt= k[SC]

d[C]

dt= −a[S][C] + (d + k)[SC] = −

d[SC]

dt

⎧

⎨ ⎪

⎩ ⎪

€

d[SC]

dt= 0⇒ [SC] =

a

d + k[S][C] = E0 −[C]

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Michaelis-Menten Kinetics (2)

• Solve for where • Therefore

where v=kE0

• If then hence

€

[C] =E0

1+ [S]/KM

€

KM = (d + k) /a

€

d[P]

dt= k[SC] =

k[S][C]

KM=

k[S]E0

KM + [S]=

v[S]

KM + [S]

€

[SC]/ E0 <<1

€

E0 ≈ [C]

€

d[P]

dt≈

k[S][C]

KM + [S]

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Michaelis-Menten in Cellerator

€

S ⇒ P (literally, {S ⇒ P, MM[K,v]})

means d[P]

dt=

v[S]

K + [S]= −

d[S]

dt

€

S ⇒ PC

(literally, {S ⇒ PC

, MM[K,v]})

means d[P]

dt=

v[C][S]

K + [S]= −

d[S]

dt

€

S ⇒ PC

(literally, {S ⇒ PC

, MM[a ,d, k]})

means d[P]

dt=

k[C][S]

(k + d) /a + [S]= −

d[S]

dt

19

Comparison of models

20

GTP: A molecular switch

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GTP: A reaction schema

€

R + LE RL

RL + (αGDP)(βγ ) → (RL)(αGDP)(βγ )

(RL)(αGDP)(βγ ) → (RL)(αβγ ) + GDP

(RL)(αβγ ) + (GTP) → (RL)(αGTP)(βγ )

(RL)(αGTP)(βγ ) → (RL) +(αGTP)(βγ )

(αGTP)(βγ ) → (αGTP) + (βγ )

αGTP + X → αGDP + X * +P

αGTP + X → αGTP + X *

βγ + Y → βγ + Y*

αGTP → αGDP + P

αGDP + βγ → (αGDP)(βγ )

R *+(αGDP)(βγ ) → (R*)(αGDP)(βγ )

22

GTP: Cellerator schema

23

GTP: Cellerator Simulation

24

RASGTP Switch

25

Cascades

26

MAPK: Mitogen Activate Protein Kinase

Cell Growth and Survival

Heat Shock, Radiation, Chemical, Inflamatory

Stress

lab of Jim Woodget, http://kinase.uhnres.utoronto.ca/

27

MAPK Cascade

€

KE K∗E K∗∗

Ph3

KK∗∗

€

KKE KK∗E KK∗∗

Ph2

KKK∗

€

KKKE KKK∗

Ph1

S

Reactions in solution (no scaffold)

28

MAPK in Solution Kinase Reactions

1st Stage

KKK+SKKK-SKKK-S KKK+SKKK-SKKK*+S

2nd Stage (1st Phosphate group)

KK+KKK*KK-KKK*

KK-KKK*KK+KKK*

KK-KKK*KKK*+KK*

2nd Stage (2nd Phosphate group)

KKK*+KK*KK*-KKK*

KK*-KKK*KKK*+KK*

KK*-KKK*KKK*+KK**

3rd Stage (1st Phosphate group)

K+KK**K-KK**

K-KK**K+KK**

K-KK**KK**+K*

3rd Stage (2nd Phosphate group)

KK**+K*K*-KK**

K*-KK**KK**+K*

K*-KK**KK**+K**

Phosphatase Reactions

1st Stage

KKK*+Ph1KKK*-Ph1

KKK*-Ph1 KKK+ Ph1

KKK*- Ph1 KKK*+ Ph1

2nd Stage (1st Phosphate group)

KK*+Ph2KK*-Ph2

KK*- Ph2 KK+ Ph2

KK*- Ph2 KK*+ Ph2

2nd Stage (2nd Phosphate group)

KK**+Ph2KK**-Ph2

KK**- Ph2 KK*+ Ph2

KK**- Ph2 KK**+ Ph2

3rd Stage (1st Phosphate group)

K*+Ph3K*-Ph3

K*- Ph3 K+ Ph3

K*- Ph3 K*+ Ph3

3rd Stage (2nd Phosphate group)

K**+Ph3K**-Ph3

K**- Ph3 K*+ Ph3

K**- Ph3 K**+ Ph3

29

MAPK Cascade on Scaffold

• Scaffold binding significantly increases the rate of phosphorylation

• Scaffold has 3 slots: one for each kinase• Each slot can be in different states

– Slot 1: empty, KKK, or KKK* bound– Slot 2: empty, KK, KK*, or KK** bound– Slot 3: empty, K, K*, K** bound

• Enter/leave scaffold in any order• KKK* and either KK or KK* must be bound at same

time produce KK**, etc.• Number of reactions increases exponentially with

number of slots

30

Effect of Scaffold on SimulationsNumber of Reactions

10

100

1000

10000

100000

2 3 4 5 6

Number of slots (N)

Single Phosphorylation

Double Phosphorylation

31

Reactions in MAP Kinase Cascade

• Phosphorylation in Solution

• Binding to Scaffold

• Phosphorylation in Scaffold

Kij ⇔

Phi

Ki+1ai+1

Kij+1 i =1,K ,n−1, j =0,K ,aj −1

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪

Sp1,L ,pi=ε,L ,pn +Ki

j ↔ Sp1,L ,pi=j,L ,pn{ }, pi =ε,0,1,K ,ai, i ≠ j

0,1,K ,ai, i = j

⎧ ⎨ ⎩

Sp1,L ,pi−1=j<ai−1,pi=ai ,L ,pn+K ⇔ Sp1,L ,pi−1=j+1,pi=ai,L ,pn{ }

Sp1,L ,pi−1=j<ai−1,pi=ai ,L ,pn → Sp1,L ,pi−1=j+1,pi=ai ,L ,pn{ }

32

Effect of Scaffold on MAPK00.10.20.30.40.50.60.70.800.511.52 Scaffold ConcentrationControl*K4 - K3 complex formationPhosphatases in scaffoldEqual rate constants*** Control Simulation: k2/k1=1000, no phosphatases, K4 does not form complex

** k2/k1=1 (rate constants for phosphorylation steps of MAPK and MAPKKK)MAPK**0100∫dt

33

Stochastic Comments

• When the number of molecules is small the continuous approach is unrealistic– Differential equations describe probabilities and not

concentrations

• At intermediate concentrations the continuous approach has some validity but there will still be noise due to stochastic effects.

– Langevin Approach:

€

A → Bk

⇒dP[b(t)]

dt= kP[a(t)]

€

d[B]dt

= k[A]+ f (t)

34

Direct Stochastic Algorithm

Gillespie Algorithm (1/3):• At any given time, determine which reaction is

going to occur next, and modify numbers of molecules accordingly

€

Reactions : R1,R2,...,RN

Rate Constants : k1,k2,...,kn

Concentrations : X1,X2,..., XM in volume V

Numbers : Ni = XiV

State of system at t : {X1,X2,...,XM }

# of Distinct Molecular Combinations in Ri : h1,h2,...,hN

hi depends combinatorically on the X1,X2,...,XM Gillespie DT (1977) J. Phys. Chem. 81: 2340-2361.

35

Gillespie Algorithm (2/3)

€

Probability that reaction Ri will occur in (t, t + dt) :

Pi (t)dt = aie−t a jj∑

dt, ai = hiki

Probability that Ri is the next reaction :

Pi = Pi (t)dt∫ = ai (a1 + a2 +L + aN )

Probability that some reaction will occur in (t, t + dt) :

P(t)dt = Pi (t)i∑ dt = aii∑( )e−t a jj∑

dt

36

Gillespie Algorithm (3/3)

Let t=0

While t<tmax {

Calculate all the ai=hiki and a0=aj

Generate two random numbers r1, r2 on (0, 1)

The time until the next reaction is =(1/a0)ln(1/r1)Set t = t + Reaction Rj occurs at t, where j satisfies

a1+a2+…+aj-1 < r2a0 ≤ aj+aj+1+…+an

Update the X1,X2,…,Xn to reflect the occurance of reaction Rj

}

37

Stochastic MAPK Simulation (1/3)

38

Stochastic MAPK Simulation (2/3)

39

Stochastic MAPK Simulation (3/3)

40

Analysis of Multi-step reactions

€

X + nL Ed

aZ

€

d[Z ]dt

= a[X][L]n − d[Z ] = 0

[X]SS =d[Z ]SS

a[L]n = N −[Z ]SS

N =[X]+[Z ]

[Z ]SS =N[L]n

[L]n +(d /a)

Steady State Solution

Adding steps increases sensitivity

€

X + L Ed

aXL, XL + L E

d

aXLL, XLL + L E

d

aXLLL, L XLn−1 + L E

d

aXLn

Simplify to:

41

Analysis of multi-stage reactions

• Consider two stages of a cascade with m and n steps

• Steady State:

€

Y =ax[X]m

Kx +[X]m , Z =ay[Y]n

Ky +[Y]n

€

Z =

ayax[X]m

Kx +[X]m

⎡

⎣ ⎢

⎤

⎦ ⎥

n

Ky +ax[X]m

Kx +[X]m

⎡

⎣ ⎢

⎤

⎦ ⎥

n =ayax

n[X]mn

Ky Kx +[X]m( )

n+ ax

n[X]mn

42

Analysis of multi-stage reactions

• If [X]<<Kx

• Hill exponent is product of m and n– E.g., a three-step stage followed by a four-step stage behaves like a

12-step stage

• By incorporating negative feedback can produce high-gain amplification (see refs).

€

Y =ax[X]m

Kx +[X]m ≈ax

Kx[X]m

⇒ Z =ay[Y]n

Ky +[Y]n =

ayax

Kx[X]m ⎛

⎝ ⎜

⎞

⎠ ⎟n

Ky +ax

Kx[X]m ⎛

⎝ ⎜

⎞

⎠ ⎟n =

ay[X]mn

KyKxn /ax

n +[X]mn

43

Oscillators in Nature

• Where they occur (to name a few):– Circadian rhythms– Mitotic oscillations– Calcium oscillations– Glycolysis– cAMP– Hormone levels

• How they occur: feedback– Both negative & positive feedback systems– Some have feed-forward loops also

44

Negative feedback: canonical model*

€

dxdt

= S −αx − βy, dydt

= γx −δy, α ,β ,γ ,δ ≥ 0

€

′ ′ x − 2a ′ x +bx =δS, a = −(α +δ) /2 ≤ 0, b =αδ + βγ ≥ 0

Equivalent second order system:0

Characteristic equation:

€

r2 − 2ar +b = 0⇒ r = a ± a2 −b

€

a2 < b⇒ stable spiral

a = 0⇒ undamped oscillations

δS /b = steady state

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

*Hoffmann et al (2002) Science 298:1241a=0.25,b=10,=1,S=5

45

2-species ring oscillator

X X*

Y

Y*

€

X ⇒ X*Y

,Y ⇒ Y *X*

, X* ⇒ XY *

,Y * ⇒ YX

€

d[X]dt

=v3(1−[X])(1−[Y])

1+ KM 3 −[X]−

v1[X][Y]KM1 +[X]

d[Y]dt

=v4[X](1−[Y])1+ KM 4 −[Y]

−v2 (1−[X])[Y]

KM 2 +[Y]

€

X + X* =1

Y +Y * =1

46

3-species ring oscillator

X X*

Y Y*

Z Z*

€

X ⇒ X*Z

,Y ⇒ Y *X

,Z ⇒ Z*Y

, X* ⇒ XZ*

,Y * ⇒ YX*

, Z* ⇒ ZY *

€

d[X]dt

=v(1−[X])(1−[Z])

1+ KM −[X]−

v[X][Z]KM +[X]

d[Y]dt

=v(1−[X])(1−[Y])

1+ KM −[Y]−

v[X][Y]KM +[Y]

d[Z]dt

=v(1−[Y])(1−[Z])

1+ KM −[Z]−

v[Y][Z]KM +[Z]

47

3-species Ring Oscillator

v=10KM

Robust oscillations

v=KM

Damped oscillations

48

Repressilator

Z PZ

Y PY

X PX

φ

φφφ

φφ

RNA

RNA

RNA

Constructed in E. coliElowitz & Leibler, Nature 403:335 (2000)

49

Repressilator Model Simulations

€

d[X]dt

=α 0 +α +α1[PY]n

K n +[PY]n − k[X], d[PX]

dt= β{[X]−[PX]}

d[Y]dt

=α 0 +α +α1[PZ]n

K n +[PZ]n − k[Y], d[PY]

dt= β{[Y]−[PY]}

d[Z ]dt

=α 0 +α +α1[PX]n

K n +[PX]n − k[Z], d[PZ]

dt= β{[Z]−[PZ]}

50

Cell Division - Canonical Model

MI M

XI X

C φφ

Goldbeter (1991) PNAS USA, 88:9107

€

d[C]dt

= 0.023− 0.00333[C]−0.1[C][X]0.02 +[C]

d[M]dt

=0.5[C](1−[M])

(0.3+[C])(1.1−[M])−

0.167[M]0.1+[M]

d[X]dt

=0.1[M](1−[X])

1.1−[X]−

0.1[X]0.1+[X]

“Minimal” Model of Cell Division

51

Cell Division - Canonical Model

52

Multi-cellular networks

€

ddt

xij = f (x1

j ,..., xNj )+ Λjkg jk (x1

k ,..., xNk )

k∈Nbr( j)∑

+ M ijk (xik − xk

j )k∈Nbr( j)∑

Intracellular Networke.g., of mass action, etc.

Transport, ligand/receptor interactions, etc

Diffusion Tensor

Connection matrixSet of neighbors

of cell j

Species xi in cell j

53

Example - coupled oscillators

€

n Coupled Oscillators

′ ′ x i +ω2xi + α jx jj∈nbr(i)∑ = 0

i =1,2,...,n

€

Two Coupled Oscillators

′ ′ x +ω2x +αy = 0

′ ′ y +ω2y +αx = 0

Two uncoupled Oscillators Two Coupled Oscillators, =.1

54

Example - 105 coupled oscillators

55

Example - 105 coupled oscillators

QuickTime™ and aVideo decompressor

are needed to see this picture.

56

Coupled nonlinear oscillators

MI M

XI X

C φφ

MI M

XI X

C φφ

MI M

XI X

C φφ

MI M

XI X

C φφ

MI M

XI X

C φφ

MI M

XI X

C φφ

Arbitrarily let species X in CMX model diffuse to adjacent cells

57

Coupled CMX Oscillators

€

d[C j]

dt= 0.023− 0.00333[C j]−

0.1[C j][X j]

0.02 +[C j]

d[M j]

dt=

0.5[C j](1−[M j])

(0.3+[C j])(1.1−[M j])−

0.167[M j]

0.1+[M j]

d[X j]

dt=

0.1[M j](1−[X j])

1.1−[X j]−

0.1[X j]

0.1+[X j]+ D ([Xk ]−[X j])k∈Nbrs( j)∑

•All oscillating at same frequency•But different phases•What happens if you have 105 coupled oscillators with random phase shifts?

58

105 CMX Oscillators: uncoupled

59

105 CMX Oscillators: uncoupled

QuickTime™ and aVideo decompressor

are needed to see this picture.

60

105 CMX Oscillators: low coupling

€

D = 0.01

61

105 CMX Oscillators: higher coupling

€

D = 0.1

62

105 CMX Oscillators: higher coupling

€

D = 0.1

QuickTime™ and aVideo decompressor

are needed to see this picture.

63

105 CMX Oscillators: Random Period

Uncoupled motion

64

105 CMX Oscillators: Random Period

€

D = 0.1

65

105 CMX Oscillators: Random Period

QuickTime™ and aAnimation decompressor

are needed to see this picture.

QuickTime™ and aAnimation decompressor

are needed to see this picture.

Uncoupled Coupled Oscillators

66

Pattern Formation

Activator-Inhibitor Models

Single Diffusing Species

•Self-activating (locally)

•Self-inhibitory (externally)

Two Diffusing Species

•X: Activator

•Y: Inhibitor

A φ

A φ

A φ

X Y φφ

X Y φφ

67

Two species pattern formation model

Continuous model*:

€

d[X]dt

= a −b[X]−[X]2 /[Y]+ D1∇2[X]

d[Y]dt

=[X]2 −[Y]+ D2∇2[Y]

Discrete implementation:

€

{∅E X j , a, b}, {Y j →∅ , 1},

{X j → X j +Y j , X j}, {2X j → X j , 1/Y j},

{X jE Xi , D1}, {Y jE Yi , D2}

*See Murray Chapter 14 for detailed analysis

68

Single species pattern formation model

Continuous (logistic) model:

€

d

dt[A] = v[A](1−[A]/K) −

vM [A] [Anbr ]nbrs∑

[A] + KM

Discrete implementation:

€

A[i] Ev /K

v2A[i], rate constants v, v/K

A[ j]⇒ ∅A[ i ]

, Michaelis constants vM ,KM

Steady State Equation (v=K=vM=KM=1, x=[A])

€

0 = x(1− x) −x

1+ xxii∑ ⇒ x = 0 or 0 =1− x2 − xii∑

69

Single species pattern formation model

€

x = 0 or 0 =1− x2 − xii∑

€

x =1 is a steady state only if all neighbors are at x=0

Suppose that there are nx neighbors in state x and all other neighbors are in state x=0, 0≤nx6

€

0 =1− x2 − nx x ⇒ x = (1/2)(−nx + nx2 + 4)

Example: case when nx =1 (exactly one neighbor at x, all others at 0):

Question: what other combinations are possible?

€

x = (1/2)(−1+ 5) ≈ 0.618034

70

Single Species Model - 105 Cells

71

Single Species Model - 105 Cells

QuickTime™ and aVideo decompressor

are needed to see this picture.

72

Provide:•most of our food and

fiber• all of our paper, cellulose, rayon

• pharmaceuticals• feed stock

• waxes• perfumes

Shoot Apical Meristemgrowing tip of a plant

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. M. M

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Computable Plant Project

73

Computable Plant Project

• NSF (USA) Frontiers in Integrative Biological Research (FIBR) Program

• S/W Architecture: Production-scale model inference– Models formulated as cellerator reactions or SBML– C++ simulation code autogenerated from models– Mathematical framework combining transcriptional regulation,

signal transduction, and dynamical mechanical models– Simulation engine including standard numerical solvers and

plot capability– Nonlinear optimization and parameter estimation– ad hoc image processing and data mining tools

• Image Acquisition– Dedicated Zeiss LSM 510 meta upright laser scanning

confocal microscope.• http://www.computableplant.org

74

Computable Plant Project

ExperimentsData Mining

Data Sets

AutomatedCode Generation

Optimization

Simulation

Mathematical ModelGeneration

Regulations &Reactions

Online

Solvers Plotters

Local

User

75

Model Organism

Arabidopsis Thaliana

76

Cell Identification in image z-stack

77

Identification of Cell Birth

78

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. M. M

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, Cal

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Div

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Bio

logy

Shoot Apical Meristem

79

Meristem Pattern Maintenance Model

80

Simulation of Meristem Growth

QuickTime™ and aVideo decompressor

are needed to see this picture.

81

Systems Biology Markup Language

http://sbml.org

libsbml (C++)

MathSBML (Mathematica)

82

The Standard Paradigm of Biology

RNA

83

Microarrays Produce a lot of data!

Affymetrix GeneChip® microarray. Images courtesy of Affymetrix.

84

RNA Fragments are Selectively Sticky

85

Affymetrix GeneChip® Scanner 3000 with workstation

Data from an experiment showing the expression of thousands of genes on a single GeneChip® probe array. Images courtesy of Affymetrix.

86

Model Inference: Fitting A Model to Data

• Cluster to reduce data size

• Use simplest possible mathematical possible to determine connectivity– Fit parameters with some optimization

process: simulated annealing, least squares, steepest descent, etc.

– Refine model with biological knowledge– Refine with better accurate math model– … and repeat until done …

87

Clustering

Time

Ge

ne

88

Data clusters in two dimensions

x

y

Concentration at time t1

Con

cent

rati

on a

t ti

me

t 2

Plot ([X[t3 1],X[t2],X[t3],…,X[tn]) for every species

89

Data clusters in two dimensions

x

y

Concentration at time t1

Con

cent

rati

on a

t ti

me

t 2

90

Clusters (may) correspond to functional modules

4: Input0: Output

Signal Transduction Network

1

2

3€

T00 T01 T02 T03 T04

T10 T11 T12 T13 T14

T20 T21 T22 T23 T24

T30 T31 T32 T33 T34

T40 T41 T42 T43 T44

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟

91

Approximation Models

• Linear

• S-Systems (Savageau)

• Generalized Mass Action

€

d[X]idt

= M ij[X] jj=1

n

∑

€

id[X]i

dt= kia

+ [X] jciaj

+

j∏a∑ − kia− [X] j

ciaj−

j∏a∑€

id[X]i

dt= ki

+ [X] jcij

+

j∏ − ki− [X] j

cij−

j∏

92

Approximation Models

• Generalized Continuous Sigma-Pi Networks

€

id[X]i

dt= g(ui + h j )

ui = ˆ M ijk [X] j[X]kk∑j∑ + M jk [X] jj∑

g(x) =12

1+x

1+ x2

⎛

⎝ ⎜

⎞

⎠ ⎟

93

Approximation Models

• Recurrent Artificial Neural Networks

• Recurrent Artificial Neural Networks with controlled degradation€

id[X]i

dt= g(ui + hi ) − λ i[X]i , ui = M ij[X] jj∑

€

id[X]i

dt= g(ui + hi ) − λ i[X]i g( ˆ u i + ˆ h i )

ui = M ij[X] jj∑ , ˆ u i = ˆ M ij[X] jj∑

94

Approximation Models

• Recurrent Artificial Neural Networks with biochemical knowledge about some species

€

id[X]i

dt= g(ui + hi ) − λ i[X]i + τ i

d[X]idt Cellerator

ui = M ij[X] jj∑

Known or hypothesized interactions due to mass action, Michaelis-Menten, or other reactions (A priori knowledge or assumptions)

95

Approximation Models

• Multicellular Artificial Neural Networks with biochemical knowledge about some species:

€

ad[X]a

i

dt= g(ua

i + ha ) − λ a[X]ai +γa[X]a

i +δa[X]a,exti

+τ id[X]i

dt Cellerator

€

uai = Mab[X]b

ib∑

+ Λijj∑ ˆ M ab[X]b

jb∑ + [X]c

i ˜ M ac(1) ˜ M cb

(2)[X]bj

b∑c∑( )

Resources

Diffusion

Lower index: species; Upper Index: CellGeometric

Connections

96

Stripe Formation in Drosophila

J Exp Zoology 271:47-56

Dashes- ObservationsSolid - Model

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, JT

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/ana

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philaObserved

Reinitz, Sharp, MjolsnessExper. Zoo. 271:47-56 (1995)

97

Some Important Meetings

• ISMB-2004, Scotland, ≈ 30 July 04Intelligent Systems in Molecular Biology

2003: Australia; 2005: US; 2006:Brazil

• ICSB-2004, Heidelberg, Oct 04International Conference on Systems Biology

SBML Forum held as satellite meeting

2003:US; 2002:Sweden; 2001:US; 2000: Japan

• PSB-2005, Hawaii, Jan 05 Pacific Symposium on Biocomputing

• RECOMB, Spring 05 Research in Computational Molecular Biology

• BGRS-04, July 04, Semiannually in NovosibirskBioinformatics of Genome Regulation and Structure

• Satellite meetings of many major biology and computer science meetings: SIAM, ACB, IEEE, ASCB (US), Neuroscience, IBRO,..

98

Collaborators

• CelleratorCellerator– Eric Mjolsness, U. California, Irvine (Computer )Eric Mjolsness, U. California, Irvine (Computer )

– Andre Levchenko, Johns Hopkins (Bioengineering)Andre Levchenko, Johns Hopkins (Bioengineering)

• Computable Plant - Eric Mjolsness, PIComputable Plant - Eric Mjolsness, PI– Elliot Meyerowitz, Caltech (Biology)Elliot Meyerowitz, Caltech (Biology)

– Venu Reddy, Caltech (Biology)Venu Reddy, Caltech (Biology)

– Marcus Heisler, Caltech (Biology)Marcus Heisler, Caltech (Biology)

– Henrik Jonsson, Lund, Sweden (Physics)Henrik Jonsson, Lund, Sweden (Physics)

– Victoria Gor, JPL (Machine Learning)Victoria Gor, JPL (Machine Learning)

• SBML (John Doyle, PI, Caltech; H. Kitano, Japan)SBML (John Doyle, PI, Caltech; H. Kitano, Japan)

– Mike Hucka, Caltech (Control & Dynamical Systems)Mike Hucka, Caltech (Control & Dynamical Systems)

– Andrew Finney, University of Hertfordshire, UKAndrew Finney, University of Hertfordshire, UK