Part 3: Introduction to Master Equation Cells: and Complex Initial Conditions in Lattice re cells Microbes en Biophysics, and UC urgh, June 6-8, 2016 rson Joseph R. Peterson and Michael J. Hallock Luthey-Schulten Laboratory University of Illinois at Urbana-Champaign NIH Workshop on Computational Biophysics November 17th, 2016 George Institute of Technology Atlanta, GA, USA
Introduction to Master Equation and LM Simulations Theory Master Equation Chemical Master Equation Stochastic Gene Expression Predator-Prey Model Effect of DNA Replication Reaction-Diffusion Master Equation Min System Creating Complex Geometries in Lattice Microbes
Biological Modeling at Different Scales Atoms Subcellular structure Single cells Multicellular organisms & populations 1Å 1nm 10nm 100nm 1µm 10µm 100µm 1mm 1cm 10cm Length scale CG Timesteps Femtoseconds Picoseconds Microseconds ns µs ms s hr Molecules to Macromolecular assemblies Whole Cells and Colonies
Master Equation Deterministic ODE Probabilistic ODE d~x dt = A~x dp (~s, t) dt = AP (~s, t) State Vector ~s = 0 B @ s 1 s 2 1 C A Transition Matrix (Probability of Birth/Death) A(t) =A(0) s N
Chemical Master Equation dp (~s, t) dt = AP (~s, t) A = R S Transition Matrix Represents Chemical Reactions by: Rate Matrix 0 1 T R= B @ 1 2 M C A S = 0 B @ Stoichiometric Matrix s 1,1 s 1,2 s,m............ s 1,M s N,M 1 C A where: and: i s i,j are propensities for a reaction to occur are the counts of each reactant
Chemical Master Equation dp ( ~ s, t) dt = NX µ=1 µ (~s s µ )P (~s s µ,t) µ (~s)p (~s, t) Transition propensities are related to macroscopic rate constants: First order: s i k! pi (~s) =ks i Second order: s i + s j k! pi (~s) = k V s is j Probability of reaction µ occurring in some time is: µ (~s)dt
Chemical Master Equation dp ( ~ s, t) dt = NX µ=1 µ (~s s µ )P (~s s µ,t) µ (~s)p (~s, t) s-3 s-2 s-1 s s+1 s+2 s+3
Chemical Master Equation dp ( ~ s, t) dt = NX µ=1 µ (~s s µ )P (~s s µ,t) µ (~s)p (~s, t) t + dt t + dt s-3 s-2 s-1 s s+1 s+2 s+3
Chemical Master Equation dp ( ~ s, t) dt = NX µ=1 µ (~s s µ )P (~s s µ,t) µ (~s)p (~s, t) t + dt t + dt s-3 s-2 s-1 s s+1 s+2 s+3
Stochastic Gene Expression Reaction Scheme D k t! D + m M k d! ; Constitutive Gene Expression dm ODE dt = k t k d m Chemical Master Equation dp (m, t) dt = k t P (m, t) k d mp (m, t) + k t P (m 1,t)+k d (m + 1)P (m +1,t)
Stochastic Gene Expression
Stochastic Gene Expression
Stochastic Gene Expression
Stochastic Gene Expression 100 Replicates
Exact Solution to Constitutive Gene Expression Moment Generating F (z,t) = @F (z,t) @t Function = 1X m=0 1X m=0 z m P (m, t) mz m 1 P (m, t) dp (m, t) dt CME = k t P (m, t) k d mp (m, t) + k t P (m 1,t)+k d (m + 1)P (m +1,t) Multiply by z m and sum over m @F (z,t) @t @F (z,t) @t = k t 1 X m=0 + k t 1 X m=0 z m P (m, t) k d 1 X m=0 z m P (m 1,t)+k d 1 X Factor out z and Compute Moments = k t F (z,t) k d zf(z,t) + k t zf(z,t)+k d @F (z,t) @t mz m P (m, t) m=0 z m (m + 1)P (m +1,t)
Exact Solution to Constitutive Gene Expression @F (z,t) @t = k t F (z,t) k d zf(z,t) + k t zf(z,t)+k d @F (z,t) @t Assume Steady-State and Solve F (z,t) =Ce k t k d z Taylor Expand and Group Terms F (z) =e k t k d 1 X m=0 m k t k d m! z m P (m) = e k t k d k t k d m m!
Exact Solution to Constitutive Gene Expression P (m) = e k t k d k t k d m m! hmi = k t k d Var[m] = k t k d Exact Distribution Simulated Distribution
Lotka-Volterra (Predator-Prey) Model ODEs Prey k a " " X Predator dx dt = k a k p XY X +Y Y k p " " 2Y k d " " dy dt = k pxy k d Y
Lotka-Volterra (Predator-Prey) Deterministic Solution
Lotka-Volterra Simulations Stable Limit Cycle Predator Extinction Event
Effects of DNA Replication on mrna Noise [2] Replication Schematically ori Replication Model [1] 40 Minute Doubling Time Cell Division 12.5 30 replication fork Rounds of Replication DNA Replication Fork 0min 10 15 40 50 55 Gene 2 80min Gene 1 ter t D 45min ori ter Reaction System I Example Simulation II III D(t) k t! D(t)+m m k d!? 1) Cooper and Helmstetter, (1968) J. Mol. Biol. 31:519. 2) J.R. Peterson, J.A. Cole, J. Fei, T. Ha, Z. Luthey-Schulten, (2015) PNAS, 112:15886. t r t d Time (min) 200cells t D = 70min t r = 25min
Exact Solution Constitutively Expressed Genes are Poisson-Distributed throughout the cell cycle ( kt 2 m(t) = m(t) = k d 0 <t<t r k t k d 2 e k d(t t r ) t r <t<t D Time-averaging CME yields moments of apple hmi = hmi 1 1+f + e fk dt D 1 k d t D Var[m] =hmi hmi 2 + hmi 2 1 Fano[m] =1 hmi + hmi2 1 hmi P (~s) f = t D apple 1+3f + 8e fk dt D e 2fk dt D 7 2k d t D apple 1+3f + 8e fk dt D e 2fk dt D 7 2k d t D t D t r Only mean, degradation rate and doubling time are needed
Model Validation Simulations Experiments [1] t D = 40min hmi = 10 f =0.35 k d =0.126min 1 1 Ignoring mrna dynamics gives large errors Analytical theory nearly exact 1) Jones et al, (2014) Science, 346:1533.
Region of importance Error in noise when neglecting mrna relaxation* 0.6 Mean mrna cspe ptsg hfq atph eno cyoc ompc 0.4 0.2 0-0.2 = F sim F TI F TI -0.4-0.6 0 2 4 6 8 10 12 14 Half-Life (min) Effect is large when: mrna half-life is large mean mrna count is high * 70 min doubling time
Other Noise Sources Regulated Gene expression RNA polymerase variability (<m>/10 Fano units) Relaxing assumption that mrna relaxes before cell division
Conflicting Interpretation of Experiments 4 3.5 Transcriptional regulation Gene duplication RNAP variability Potential Misinterpretation Fano factor contribution 3 2.5 2 of Experimental Data Parameters t D = 40min 1.5 hmi = 10 1 f =0.35 1 No Gene Duplication Time Independent Model Time Dependent Model k d =0.126min 1
Reaction-Diffusion Master Equation dp (~s, t) dt = AP (~s, t) A = R S + D
Reaction-Diffusion Master Equation dp (~s, t) dt = AP (~s, t) A = R S + D Physical Space x-1 x x+1
Reaction-Diffusion Master Equation dp (~s, t) dt = AP (~s, t) A = R S + D d i 2 x-1 x x+1
Reaction-Diffusion Master Equation dp (~s, t) dt = AP (~s, t) A = R S + D d i 2 x-1 x x+1
Min System Example Sets up the center plane for cell division https://www.youtube.com/watch?v=6bnb9oeh3ta https://en.wikipedia.org/wiki/min_system#/media/file:mincde_system2.svg
Min System Example Schematic Reaction System Cytoplasm MinDADP k1 k5 MinDEm MinDATP MinE k4 k2 MinDm k3 MinDm MinDm Extracellular Space J.R. Peterson, M.J. Hallock, J.A. Cole, Z. Luthey-Schulten, (2013) PyHPC '13: Proceedings of the 3rd Workshop on Python for High-Performance and Scientific Computing.
Effect of in vivo crowding on repressor re-binding Complex Geometries in Lattice Microbes Examining Propensity for a Transcription Factor to Rebind to DNA after Unbinding as a Function of Packing Fraction E. Roberts, J. Stone,L. Sepulveda, W.M. Hwu, ZLS, IEEE, 2009
Complex Geometries in Lattice Microbes
Complex Geometries in Lattice Microbes 4 programmatically generated epithelial cells https://s-media-cacheak0.pinimg.com/originals/7f/ c4/ed/ 7fc4ed17cfa66eb820a99d6fbf 3055fc.jpg
Acknowledgements Collaborators Luthey-Schulten Laboratory Funding and Computer Time