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BioSimWare: A P Systems-based Simulation Environment for Biological Systems Daniela Besozzi 1 , Paolo Cazzaniga 2 , Giancarlo Mauri 2 , Dario Pescini 2 Universit` a degli Studi di Milano Dipartimento di Informatica e Comunicazione Via Comelico

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SLIDE 1

BioSimWare: A P Systems-based Simulation Environment for Biological Systems

Daniela Besozzi1, Paolo Cazzaniga2, Giancarlo Mauri2, Dario Pescini2

Universit` a degli Studi di Milano Dipartimento di Informatica e Comunicazione Via Comelico 39, 20135 Milano, Italy besozzi@dico.unimi.it Universit` a degli Studi di Milano-Bicocca Dipartimento di Informatica, Sistemistica e Comunicazione Viale Sarca 336, 20126 Milano, Italy cazzaniga/mauri/pescini@disco.unimib.it

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SLIDE 2

Outline

1

BioSimWare

2

Stochastic Simulations Algorithms for Single and Multi-Volume Systems

3

Tools for The Analysis of Stochastic Simulations Parameter Estimation (PE) Parameter sweep applications (PSA)

4

Applications The Schl¨

  • gl System

The Brussellator Stiff Systems Bacterial Chemotaxis Simulation of Fredkin Circuits by Chemical Reaction Systems

5

Conclusion

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SLIDE 3

Outline

1

BioSimWare

2

Stochastic Simulations Algorithms for Single and Multi-Volume Systems

3

Tools for The Analysis of Stochastic Simulations Parameter Estimation (PE) Parameter sweep applications (PSA)

4

Applications The Schl¨

  • gl System

The Brussellator Stiff Systems Bacterial Chemotaxis Simulation of Fredkin Circuits by Chemical Reaction Systems

5

Conclusion

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SLIDE 4

Introduction

BioSimWare is a simulation environment based on P systems, that provides a user-friendly framework for the modeling of biological systems ranging from cellular processes to population phenomena.

1 2 3

substances (objects) reactions (evolution rules)

  • rganelles (regions)

membrane

a a a a c c c b b b a c a a a b b c c c c'→(c, in2) a+c→(b, out) a+b→(c, here) c→(c', out) b+c→(a, here) a→(a, out)

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SLIDE 5

User Interface: rules specification

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SLIDE 6

User Interface: system conditions specification

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SLIDE 7

User Interface: plot of the dynamics

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SLIDE 8

Outline

1

BioSimWare

2

Stochastic Simulations Algorithms for Single and Multi-Volume Systems

3

Tools for The Analysis of Stochastic Simulations Parameter Estimation (PE) Parameter sweep applications (PSA)

4

Applications The Schl¨

  • gl System

The Brussellator Stiff Systems Bacterial Chemotaxis Simulation of Fredkin Circuits by Chemical Reaction Systems

5

Conclusion

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SLIDE 9

Available simualtion algorithm

Single Volume SSA tau leaping adaptive tau leaping average tau leaping dynamics ODE integrator Multi Volume DPP tau-DPP Stau-DPP

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SLIDE 10

Outline

1

BioSimWare

2

Stochastic Simulations Algorithms for Single and Multi-Volume Systems

3

Tools for The Analysis of Stochastic Simulations Parameter Estimation (PE) Parameter sweep applications (PSA)

4

Applications The Schl¨

  • gl System

The Brussellator Stiff Systems Bacterial Chemotaxis Simulation of Fredkin Circuits by Chemical Reaction Systems

5

Conclusion

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SLIDE 11

PE: the goal

We would like to reproduce the target dynamics via stochastic simulations:

560 580 600 620 640 660 680 700 720 740 760 0.5 1 1.5 2 2.5 3 3.5 molecules time [a.u.] target generated

not necessarily using the same set of constants.

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SLIDE 12

PE: the problem

Quantify the “distance” among the dynamics and find the closest one

560 580 600 620 640 660 680 700 720 740 760 0.5 1 1.5 2 2.5 3 3.5 molecules time[a.u.] target 0.00245015 0.0015 0.001 0.0005 0.0001

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SLIDE 13

The fitness: Standard Distance

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SLIDE 14

The fitness: tricks

Facts related to intrinsic noise that should not be neglected: each outcome j is quantitatively different {X(τi)}j

not evenly sampled {τi}j not same number of points {= X(τi)}j

in the thermodynamic limit {X(τi)} → [X](ti)

exploit ensemble behavior (may flatten oscillations, not in phase outcome) F (X(τi))

?

= F ( X(τi) )

same parameters, possibly (almost always), generate different values of the fitness function (“weak convergence”)

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SLIDE 15

The fitness: “Area” distance

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SLIDE 16

Reconstructed dynamics

1100 1150 1200 1250 1300 1350 1400 1450 1500 0.05 0.1 0.15 0.2 CheYp Molecules Time [a.u.] HC PSO Target

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SLIDE 17

Parameter sweep applications (PSA)

A PSA consists in a repeated execution of an application (usually performed a large number of times), where each execution is achieved using a different parametrisation. In the context of biochemical stochastic models a PSA could be viewed as PSA = (P; D; M), where P = (p1; . . . ; pn) is the set of parametrisations with pi = (xi

0; ci)

D = (d1; . . . ; dn) di = (xi(0); . . . ; xi(t)) where t is the halting time of the simulation i M is the biochemical model that provides the map pi → di

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SLIDE 18

PSA: EGEE Grid

EGEE project infrastructure, a wide area grid platform for scientific applications, com- posed of thousands of CPUs, which im- plements the Virtual Organisation (VO) paradigm. More than 90 partners in 32 countries, organised in 13 Federations A Grid infrastructure spanning almost 240 sites across 45 countries An infrastructure of 41,000 CPU available to users 24 hours a day, 7 days a week More than 5 Petabytes (5 million Gigabytes) of storage Sustained and regular workloads of 30K jobs/day, reaching up to 98K jobs/day

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SLIDE 19

Perturbed parameters fitness

10 20 30 40 50 60 0 10 20 30 40 50 60 70 80 90 100 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 fitness ci ci perturbation fitness 5e-05 0.0001 0.00015 0.0002 0.00025 0.0003 5e-05 0.0001 0.00015 0.0002 0.00025 0.0003 0.00035 0.0004 0.00045 0.0005 2000 2500 3000 3500 4000 4500 5000 5500 6000 6500 fitness fitness

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SLIDE 20

Outline

1

BioSimWare

2

Stochastic Simulations Algorithms for Single and Multi-Volume Systems

3

Tools for The Analysis of Stochastic Simulations Parameter Estimation (PE) Parameter sweep applications (PSA)

4

Applications The Schl¨

  • gl System

The Brussellator Stiff Systems Bacterial Chemotaxis Simulation of Fredkin Circuits by Chemical Reaction Systems

5

Conclusion

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SLIDE 21

The Schl¨

  • gl System

r1 : A + 2X

3·10−7

− → 3X r2 : 3X

10−4

− → A + 2X r3 : B

10−3

− → 3X r4 : X

3.5

− → B

A = 1·105 B = 2·105 X = 250

100 200 300 400 500 600 700 5 10 15 20 X Molecules Time [a.u.] 100 200 300 400 500 600 700 800 50000 100000 150000 200000 250000 300000 350000 400000 X Molecules Time [a.u.] 100000 200000 300000 400000 500000 600000 700000 800000 900000 1e+06 100 200 300 400 500 600 700 800 Frequency X Molecules

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SLIDE 22

The Brusselator

r1 :A

1

− → X r2 :B + X

5·10−3

− → Y r3 :2X + Y

2.5·10−5

− → 3X r4 :X

1.5

− → λ

A = X = 200 B = 600 Y = 300

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SLIDE 23

Comparison GA & PSO: Oscillating Brusselator

5000 10000 15000 20000 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Average Best Fitness Fitness Evaluations

PSO1 PSO2 GA

500 1000 1500 2000 2500 5 10 15 20 25

Molecules Time [a.u.]

X Y X target Y target

0.2 0.4 0.6 0.8 1 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Successful Runs Fitness Evaluations

PSO1 PSO2 GA

PSO1/2 best performers GA worst average fitnesses dynamics faithfully reconstructed

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SLIDE 24

Comparison GA & PSO: Damped Brusselator

400 600 800 1000 1200 1400 1600 1800 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Average Best Fitness Fitness Evaluations

PSO1 PSO2 GA

50 100 150 200 250 300 350 400 5 10 15 20 25

Molecules Time [a.u.]

X Y X target Y target

10 20 30 40 50 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Successful Runs Fitness Evaluations

PSO1 PSO2 GA

PSO1/2 best performers GA worst average fitnesses dynamics unfaithfully reconstructed

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SLIDE 25

Stiff systems: an example

A system is said to be stiff if is characterized by well-separeted fast and slow dynamical modes, the fastest of which is stable. The decaying dimerization model: r1 :S1

1

− → λ r2 :S1 + S1

10

− → S2 r3 :S2

103

− → S1 + S1 r4 :S2

0.1

− → S3 S1 = 104 S2 = S3 = 0

SSA tau leaping adaptive tau leaping average number of steps 2.46 · 107 1 · 106 945 total execution time 175h 1m 11h 47m 1m 51s

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SLIDE 26

Decaying dimerization: algorithms comparison

1000 2000 3000 4000 5000 0.5 1 1.5 2 2.5 3 3.5 4 Molecules Time [a.u.] S1 S2 S3 1000 2000 3000 4000 5000 0.5 1 1.5 2 2.5 3 3.5 4 Molecules Time [a.u.] S1 S2 S3 1000 2000 3000 4000 5000 0.5 1 1.5 2 2.5 3 3.5 4 Molecules Time [a.u.] S1 S2 S3 1000 2000 3000 4000 5000 0.5 1 1.5 2 2.5 3 3.5 4 Molecules Time [a.u.] S1 S2 S3 SSA tau-leaping adaptive tau-leaping

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SLIDE 27

Bacterial chemotaxis

Chemotaxis allows bacteria to respond to ligand concentration gradients in their surroundings: random walk through an homogeneous environment → high switching frequency of flagellar rotation directional swimming in presence of ligand concentration gradient → reduced switching frequency of flagellar rotation adaptation: if ligand concentration remains constant → switching frequency is reset to random walk level

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SLIDE 28

Sensing Responding Adapting

Reagents Products

  • Methyl. state

1 2MCPm + 2CheW 2MCPm ::2CheW m = 0 2 2MCPm ::2CheW 2MCPm + 2CheW m = 0 3 2MCPm ::2CheW + 2CheA 2MCPm ::2CheW::2CheA m = 0 4 2MCPm ::2CheW::2CheA 2MCPm ::2CheW + 2CheA m = 0 5-8 2MCP m ::2CheW::2CheA + CheR 2MCPm+1::2CheW::2CheA + CheR m = 0, . . . , 3 9-12 2MCPm ::2CheW::2CheA + CheBp 2MCPm−1::2CheW::2CheA + CheBp m = 1, . . . , 4 13-17 2MCPm ::2CheW::2CheA + ATP 2MCPm ::2CheW::2CheAp m = 0, . . . , 4 18-22 2MCPm ::2CheW::2CheAp + CheY 2MCPm ::2CheW::2CheA + CheYp m = 0, . . . , 4 23-27 2MCPm ::2CheW::2CheAp + CheB 2MCPm ::2CheW::2CheA + CheBp m = 0, . . . , 4 28-32 lig + 2MCPm ::2CheW::2CheA lig::2MCPm ::2CheW::2CheA m = 0, . . . , 4 33-37 lig::2MCPm ::2CheW::2CheA lig + 2MCPm ::2CheW::2CheA m = 0, . . . , 4 38-41 lig::2MCPm ::2CheW::2CheA + CheR lig::2MCPm+1::2CheW::2CheA + CheR m = 0, . . . , 3 42-45 lig::2MCPm ::2CheW::2CheA + CheBp lig::2MCPm−1::2CheW::2CheA + CheBp m = 1, . . . , 4 46-50 lig::2MCPm ::2CheW::2CheA + ATP lig::2MCPm ::2CheW::2CheAp m = 0, · · · , 4 51-55 lig::2MCPm ::2CheW::2CheAp + CheY lig::2MCPm ::2CheW::2CheA + CheYp m = 0, . . . , 4 56-60 lig::2MCPm ::2CheW::2CheAp + CheB lig::2MCPm ::2CheW::2CheA + CheBp m = 0, . . . , 4 61 CheYp + CheZ CheY + CheZ 62 CheBp CheB

62 reactions, 32 molecular species

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SLIDE 29

Sensing Responding Adapting

800 1000 1200 1400 1600 1800 2000 500 1000 1500 2000 2500 3000 3500 4000 CheYp Molecules Time [sec] ODE solution Stoch solution

Sensing Responding Adapting

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SLIDE 30

CheYp and the flagellar rotation

Our assumptions: the flagellar motor switch is sensitive to a threshold level of CheYp, that is hereby evaluated as the mean value of CheYp at steady state we make a one-to-one correspondence between the behavior of a single flagellum and one temporal evolution

  • f CheYp generated by one run of the tau leaping

algorithm.

50 100 150 200 Time [sec]

μ CheYp True False CW CCW

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SLIDE 31

Many flagella and running or tumbling

T n

sync = {t ∈ ∆tsim | CCWsi(t) = true for all i = 1, . . . , n}

T n

sync is the set of all times during which all time series si are below

the threshold µ

time

s s

1 2

That corresponds to the running motion of the bacterium

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SLIDE 32

Many flagella and running or tumbling

0.5 1 1.5 2 2.5 3 3.5 2 4 6 8 10 Time [sec] Flagella < ∆ trun >

Running < ∆trun > all flagella rotate CCW

50 100 150 200 250 300 2 4 6 8 10 Time [sec] Flagella < ∆ ttumb >

Tumbling < ∆ttumb > some flagella rotate CW

70 75 80 85 90 95 100 105 2 4 6 8 10 Time [sec] Flagella < ∆ tadapt >

Adapting < ∆tadapt > length of the negative peak

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SLIDE 33

Fredkin Gate: definition

αi βi γi → αo βo γo 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 0 0 1 1 0 1 1 1 0 0 1 0 0 1 0 1 1 1 0 1 1 0 1 0 1 1 1 1 1 1 1

Reaction Constant r1 : a + b → a + d c1 = 1 · 10−3 r2 : a + c → a + e c2 = 1 · 10−3 r3 : a + B → a + D c3 = 1 · 10−3 r4 : a + C → a + E c4 = 1 · 10−3 r5 : A + b → A′ c5 = 1 · 10−3 r6 : A′ + c → A + d + e c6 = 1 · 10−1 r7 : A′ + C → A + D + e c7 = 1 · 10−1 r8 : A + B → A′′ c8 = 1 · 10−3 r9 : A′′ + c → A + d + E c9 = 1 · 10−1 r10 : A′′ + C → A + D + E c10 = 1 · 10−1 Reaction Constant r11 : a + A → λ c11 = 1 · 10−1 r12 : b + B → λ c12 = 1 · 10−1 r13 : c + C → λ c13 = 1 · 10−1 r14 : d + D → λ c14 = 1 · 10−1 r15 : e + E → λ c15 = 1 · 10−1 r16 : λ → a c16 ∈ {0, 1} r17 : λ → A c17 ∈ {0, 1} r18 : λ → b c18 ∈ {0, 1} r19 : λ → B c19 ∈ {0, 1} r20 : λ → c c20 ∈ {0, 1} r21 : λ → C c21 ∈ {0, 1}

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SLIDE 34

Fredkin Gate: definition

αi βi γi → αo βo γo a b c a d e a b C a d E a B c a D e a B C a D E A b c A d e A b C A D e A B c A d E A B C A D E

Reaction Constant r1 : a + b → a + d c1 = 1 · 10−3 r2 : a + c → a + e c2 = 1 · 10−3 r3 : a + B → a + D c3 = 1 · 10−3 r4 : a + C → a + E c4 = 1 · 10−3 r5 : A + b → A′ c5 = 1 · 10−3 r6 : A′ + c → A + d + e c6 = 1 · 10−1 r7 : A′ + C → A + D + e c7 = 1 · 10−1 r8 : A + B → A′′ c8 = 1 · 10−3 r9 : A′′ + c → A + d + E c9 = 1 · 10−1 r10 : A′′ + C → A + D + E c10 = 1 · 10−1 Reaction Constant r11 : a + A → λ c11 = 1 · 10−1 r12 : b + B → λ c12 = 1 · 10−1 r13 : c + C → λ c13 = 1 · 10−1 r14 : d + D → λ c14 = 1 · 10−1 r15 : e + E → λ c15 = 1 · 10−1 r16 : λ → a c16 ∈ {0, 1} r17 : λ → A c17 ∈ {0, 1} r18 : λ → b c18 ∈ {0, 1} r19 : λ → B c19 ∈ {0, 1} r20 : λ → c c20 ∈ {0, 1} r21 : λ → C c21 ∈ {0, 1}

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SLIDE 35

Fredkin Gate: simulations

200 400 600 800 1000 1200 1400 1600 200 400 600 800 1000 1200 1400

Molecules Time [a.u.]

a d E D

First input (αi , βi , γi ) = (0, 0, 1) at t = 0 Second input (αi , βi , γi ) = (0, 1, 1) at t = 500 a b C → a d E a B C → a D E

200 400 600 800 1000 1200 500 1000 1500 2000

Molecules Time [a.u.]

a A d e E D

First input (αi , βi , γi ) = (0, 0, 1) at t = 0 Second input (αi , βi , γi ) = (1, 0, 1) at t = 400 a b C → a d E A b C → A D e

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SLIDE 36

Fredkin Circuits

500 1000 1500 2000 2500 y1 = 1 y2 = 0 y2 = 1 200 400 600 800 1000 1200

Molecules

y3 = 0 y3 = 1 y4 = 1 y5 = 0 200 400 600 800 1000 1200 1400 1600 1800 500 1000 1500 2000

Time [a.u.]

y6 = 1 y7 = 0

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SLIDE 37

Outline

1

BioSimWare

2

Stochastic Simulations Algorithms for Single and Multi-Volume Systems

3

Tools for The Analysis of Stochastic Simulations Parameter Estimation (PE) Parameter sweep applications (PSA)

4

Applications The Schl¨

  • gl System

The Brussellator Stiff Systems Bacterial Chemotaxis Simulation of Fredkin Circuits by Chemical Reaction Systems

5

Conclusion

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SLIDE 38

Summary

BioSimWare is a simulation environments for the investigation

  • f various biological systems that can range from cellular

processes to population phenomena and to ecological systems. Features Single/Multi volume simulations (stochastic and deterministic) Parameter Estimation (HC, GA, PSO) Parameter Sweep Analysis of the dynamics Implementations Linux, Windows, Mac OS Single Processor, MPI, GRID SBML