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Models for cell and organism development ODE, PDE, and Boolean models Jan-Åke Larsson Informationskodning, ISY 11th May 2012 Proteins, proteins, proteins Essential Cell Biology, 3rd ed, Alberts et al, 2009 Examples of modeling tasks I Metabolism I Signal transduction I Gene regulation I ... (As on Wikipedia) Examples of modeling tasks I Metabolism I Signal transduction I Gene regulation I ... Nyman et al, Trends Endocr. Metabol. 23:107, 2012 Examples of modeling tasks I Metabolism I Signal transduction I Gene regulation I ... Maduro et al, Biochim Biophys Acta, 2009 Examples of modeling tasks I Metabolism I Signal transduction I Gene regulation I ... Essential Cell Biology, 3rd ed, Alberts et al, 2009 Model types pnas.org I Graph models I Dynamic models I Stochastic models Graph models pnas.org Graph models cycles point at feedback relations connected subgraphs point at functional modules pnas.org I Indicates dependence I Can handle complex networks I Dynamics not included I Statistical (Bayesian-network) methods common may reveal missing regulatory interactions provides clues about redundancy comparing different organisms can show evolutionary relations Model types pnas.org I Graph models I Dynamic models I Stochastic models Model types Nakajima et al, PLoS Comput Biol 6:e1000760, 2010 I Graph models I Dynamic models I Stochastic models Model types Nakajima et al, PLoS Comput Biol 6:e1000760, 2010 I Graph models I Dynamic models I Stochastic models Boolean models I Represented using binary vectors I Usually synchronous updating, discretized time x(t + 1) = F x(t) I Finite state space T x(t) = xhb (t), xKr (t), xpdm (t), xcas (t) 0 +1 x(t+1) = F −5 −5 0 0 0 −5 +1 0 −5 +1 0 0 x(t) −5 0 Boolean models I Represented using binary vectors I Usually synchronous updating, discretized time x(t + 1) = F x(t) I Finite state space 9 hb C D Kr 6 T x(t) = xhb (t), xKr (t), xpdm (t), xcas (t) pdm cas 3 5 7 A B E F 0 0 +1 0 x(t+1) = F −5 +1 −5 −5 0 −5 0 +1 0 0 x(t) −5 0 Boolean models I Represented using binary vectors Usually synchronous updating, discretized time x(t + 1) = F x(t) I Finite state space I Wuensche, Artificial Life Models in Software, 2nd Ed, eds. Komosinski et al, 2009 Boolean models I Can be used to study large-scale behaviour of random networks I For example, the number of attractors seem to scale as I One can also distinguish chaotic from ordered structures Wuensche, Kybernetes 32:77, 2003 √ n Generalized Boolean models I Asynchronous update I Multi-valued variables I Random update rules A Nakajima et al, PLoS Comput Biol 6:e1000760, 2010 B Model types Nakajima et al, PLoS Comput Biol 6:e1000760, 2010 I Graph models I Dynamic models I Stochastic models Model types Nakajima et al, PLoS Comput Biol 6:e1000760, 2010 I Graph models I Dynamic models I Stochastic models Continuous (ODE-) models I Time-dependent vectors in Rn I Usually continuous time (but numerics use discrete time) x0 (t) = F x(t) I The function F should be nonlinear h+ (xj , θj , m) = xjm xjm + θjm , or σ(xj , θj , m) = 1 1 + e m(θj −xj ) de Jong, J Comput Biol 9:67, 2002 Continuous (ODE-) models I Stability is more difficult to analyze I The number of attractors is hard to find I In some cases, proving that there is one is difficult I Parameter values are unknown I Usable in less complicated networks A common form is xi0 (t) = σ X ωij xj (t) − θi j This is linear close to the “thresholds” θi Continuous (ODE-) models One approach is piecewise linearization de Jong, J Comput Biol 9:67, 2002 This will simplify the analysis greatly, but also introduce spurious steady states on boundaries between the linearized regions Continuous (ODE-) models Example: dMi (t) = γM [Fi (P(t)) − Mi (t)] + ξi (t) dt dPi (t) = γP [Mi (t)) − Pi (t)] dt P [g (Si + j Jij Pj (t))]m P Fi (P(t)) = m KM + [g (Si + j Jij Pj (t))]m Nakajima et al, PLoS Comput Biol 6:e1000760, 2010 Continuous (ODE-) models Unfortunately, there are more complications. The reactions take time, resulting in a delay from presence of a transcription factor to production of the next protein in the chain x0 (t) = F (x1 (t − τ1 ), x2 (t − τ2 ), . . . , xN (t − τN ))T Ruan, in Delay Differential Equations and Applications, ed Arino et al, Springer 2006 Continuous (PDE-) models In embryonal development, position of cells (or as in Drosophila, to the right, the nucleii) also plays a role Long range signals are often mediated via diffusion of a transcription factor, leading to a PDE like ∂2x ∂x = F (x) + c · 2 ∂t ∂s This goes back to Turing (1951) Essential Cell Biology, 3rd ed, Alberts et al, 2009 Continuous (PDE-) models In embryonal development, position of cells (or as in Drosophila, to the right, the nucleii) also plays a role Long range signals are often mediated via diffusion of a transcription factor, leading to a PDE like ∂2x ∂x = F (x) + c · 2 ∂t ∂s Essential Cell Biology, 3rd ed, Alberts et al, 2009 This goes back to Turing (1951) Continuous (ODE-) models Example: dMi (t) = γM [Fi (P(t)) − Mi (t)] + ξi (t) dt dPi (t) = γP [Mi (t)) − Pi (t)] dt P [g (Si + j Jij Pj (t))]m P Fi (P(t)) = m KM + [g (Si + j Jij Pj (t))]m Nakajima et al, PLoS Comput Biol 6:e1000760, 2010 Continuous (ODE-) models ↓ Example: dMi (t) = γM [Fi (P(t)) − Mi (t)] + ξi (t) dt dPi (t) = γP [Mi (t)) − Pi (t)] dt P [g (Si + j Jij Pj (t))]m P Fi (P(t)) = m KM + [g (Si + j Jij Pj (t))]m Nakajima et al, PLoS Comput Biol 6:e1000760, 2010 Model types Nakajima et al, PLoS Comput Biol 6:e1000760, 2010 I Graph models I Dynamic models I Stochastic models Model types Nakajima et al, PLoS Comput Biol 6:e1000760, 2010 I Graph models I Dynamic models I Stochastic models Stochastic models Concentrations of substances do not vary continuously and deterministically There is gene expression noise, arising mainly in transcription Signals on the order of tens of molecules can make a difference Essential Cell Biology, 3rd ed, Alberts et al, 2009 Model: a stochastic master equation, or alternatively a stochastic differential equation Sometimes, but not always, this can be written as an ODE with a noise term: “Langevin equations” Stochastic models I good description of molecular gene regulation I requires detailed knowledge of the reaction mechanisms I . . . and reaction rates I stochastic simulation is costly On a larger time-scale, stochastic effects may level out, so that continuous and deterministic models form a good approximation Essential Cell Biology, 3rd ed, Alberts et al, 2009 Model types, briefly Graph models: Highly complex networks, no dynamics, general features, functional modules Boolean models: Moderately complex networks, coarse-grained dynamics, synergy effects can be handled Continuous models: Less complex networks, general dynamics, nonlinearity and synergy more difficult to handle Stochastic models: Used on single pathways, requires detailed knowledge, gives detailed answers Nakajima et al, PLoS Comput Biol 6:e1000760, 2010 Example: Drosophila melanogaster Example: Drosophila Example: Drosophila P 1, Pj Jij xj (t) > 0 xi (t + 1) = θi , Jij xj (t) = 0 Pj 0, j Jij xj (t) < 0 Example: Drosophila Example: Drosophila Example: Drosophila I Boolean modeling can give hints on missing parts of the network I Here, the factor X would give both wild-type and mutant behaviour I Stochastic modeling can give more insight into the stability Example: Drosophila Example: Drosophila Example: Cænorhabditis elegans I A well-studied organism I Cell lineage tree completely mapped (959 cells in adult }) I Genome completely mapped (∼ 20 000 genes) Part of lineage tree (Sulston et al, 83) Part of lineage tree (Sulston et al, 83) Boolean model for regulation I Notation: “promotor functions” g(A|B) I Synergy as logical AND in g(A|B,C) I Cell division is controlled by special promotors g(ab|Z) I Can generate all trees Part of lineage tree (Sulston et al, 83) Part of lineage tree (Sulston et al, 83) Part of lineage tree (Sulston et al, 83) Extension to discretized levels I C5 : The factor C has concentration 5 I g(h5 |C5 ): Concentration 5 of the factor h is produced, if the factor C has concentration 5 Part of lineage tree (Sulston et al, 83) Part of lineage tree (Sulston et al, 83) Another extension: external signals, for example, at cell-cell contact I I Based on physical cell proximity (morphology) — not lineage tree Two main mechanisms I I Distinguish two initially identical daughters Polarize dividing cell to generate two distinct daughters Signaling is part of the known network Maduro et al, Biochim Biophys Acta, 2009 Part of lineage tree (Sulston et al, 83) Part of lineage tree (Sulston et al, 83) Model types, briefly Graph models: Highly complex networks, no dynamics, general features, functional modules Boolean models: Moderately complex networks, coarse-grained dynamics, synergy effects can be handled Continuous models: Less complex networks, general dynamics, nonlinearity and synergy more difficult to handle Stochastic models: Used on single pathways, requires detailed knowledge, gives detailed answers Nakajima et al, PLoS Comput Biol 6:e1000760, 2010