dynamic processes in cells (a systems approach to biology)

Transcription

1 dynamic processes in cells (a systems approach to biology) jeremy gunawardena department of systems biology harvard medical school lecture 6 18 september 2014

2 weak linkage facilitates variation and evolution weak linkage allows mutations to occur within modules without compromising the organism as a whole such mutations give rise to genetic variation which is neutral or only mildly deleterious this genetic variation can proliferate and mix in the population by mating and recombination the resulting variation can provide the raw material for subsequent positive selection

3 weak linkage encourages molecular complexity gene regulation post-translational modification

4 3. time-scale separation & the linear framework

5 time-scale separation nonlinear steady state eliminate the complexity of the sub-system sub-system system fast slow E + S ES E + P

6 the linear framework nonlinear linear steady state part or all of the sub-system is represented by a graph sub-system system E + S a ES nonlinear with simple rate constants E a[s] ES linear with complex labels subject to an uncoupling condition Gunawardena, A linear framework for time-scale separation in nonlinear biochemical systems, PLoS ONE 7:e ; Mirzaev & Gunawardena, Laplacian dynamics on general graphs, Bull Math Biol 75: ; Gunawardena, Time-scale separation: Michaelis and Menten's old idea, still bearing fruit, FEBS J 281:

7 graph theory begins with euler Königsberg, modern Kaliningrad graphs are widely used as mathematical representations of biological networks Newman The structure and function of complex networks, SIAM Review 45: ; Barabasi, Oltvai, Network biology: understanding the cell's functional organization, Nature Rev Genetics 5:

8 labelled directed graphs in the linear framework a graph consists of vertices (or nodes), with edges between vertices a graph is directed if each edge has an arrow at one end, specifying a direction it is labelled if each edge has a label we shall work with graphs which are connected (in one piece, forgetting edge directions) and which have no self-loops

9 Laplacian dynamics macroscopic interpretation one-dimensional chemistry on graphs consider each edge as a chemical reaction under mass-action kinetics, with the label as the rate constant 1 a1 a3 3 a2 2 a4 Laplacian matrix system of linear ODEs Gustav Kirchhoff, Uber die Auflosung der Gleichungen, auf welche man bei der Untersuchung der linearen Verteilung galvanischer Strome gefuhrt wird, Ann Phys Chem, 72:

10 conservation law with n vertices in the graph, the Laplacian matrix is n x n since matter is neither created nor destroyed, so there is always the conservation law equivalently, the sum of the entries in each column is zero

11 Laplacian dynamics microscopic interpretation let be a time-homogeneous Markov process on the states for which infinitesimal transition rates exist define the graph,, with vertices and an edge iff give this edge the label the stochastic master equation, or Kolmogorov forward equation, for the probability of being in state i at time t, is identical to Laplacian dynamics on Mirzaev & Gunawardena, Bull Math Biol 75:

12 stability of Laplacian dynamics the eigenvalues of the Laplacian matrix are either 0 or have negative real part (*) 0 can be a repeated eigenvalue, which could give an instability but it can be shown (*) that this does not occur for any graph, G, the dynamics always tends to a stable steady state which lies in the kernel of the Laplacian (*) Mirzaev & Gunawardena, Bull Math Biol 75:

13 complexity disappears at steady state if G is strongly connected there is an unique steady state, up to a constant factor the kernel of the Laplacian is one dimensional not SC 1 strongly connected there is a directed path between any two distinct vertices a1 1 a3 3 a2 2 SC a4 a1 a3 3 a2 2 a4 despite the many degrees of freedom initially (as many as the number of vertices), once a steady state is reached, only one degree of freedom is left

14 non-strongly connected graphs any graph can be decomposed into strongly connected components (SCCs), or maximal strongly connected subgraphs the terminal SCCs (shown by *) carry the steady states of the Laplacian dynamics the terminal SCCs provide a basis for the kernel of the Laplacian the k basis vectors in the kernel are complemented by k conservation laws (*) Mirzaev & Gunawardena, Bull Math Biol 75:

15 elimination of internal variables when G is strongly connected, so that if there is a steady state then each of the can be eliminated reference node rational expressions in the symbolic labels

16 calculating 's 1. if the system reaches thermodynamic equilibrium, then detailed balance holds and can be calculated using a single path in the graph or by equilibrium statistical mechanics (next lecture) 2. if the system does not reach equilibrium and the graph is strongly connected, then can be calculated by the Matrix-Tree Theorem (this lecture). this requires many paths (or spanning trees ) in the graph. 3. if the graph is not strongly connected, then a basis for the kernel can be calculated by decomposing the graph into SCCs and using the MTT on the terminal SCCs

17 calculating the matrix-tree theorem a formula for calculating a basis element of when G is strongly connected polynomial in the symbolic labels set of spanning trees rooted at i Bill Tutte, The dissection of equilateral triangles into equilateral triangles, Proc Camb Phil Soc 44: the MTT has been independently rediscovered in biology, physics, economics, computer science, etc Mirzaev & Gunawardena, Bull Math Biol 75: see the Appendix for a proof

18 spanning trees rooted spanning tree a sub-graph T of G that 1. SPANS G every node of G is also a node of T 2. is a TREE T has no cycles, ignoring edge directions 3. is ROOTED at i i is the only node of T with no outgoing edges a graph is strongly connected if, and only if, it has a rooted spanning tree at each vertex software for enumerating spanning trees is discussed in (*) and can be found at (*) Ahsendorf, Wong, Eils, Gunawardena, A graph-based framework for modelling gene regulation reveals history-depdendent complexity away from equilibrium, submitted, 2014.

19 spanning-tree formula 1 3 a2 1 a1 1 a3 2 3 a2 3 a1 2 2 a4 a Laplacian a4 a3 3 a1 1 a4 1 a4

20 the uncoupling condition labels are allowed to be complex algebraic expressions, involving rate constants and concentrations rational function a provided that no concentration appearing in a label such as [X1] or [X2] above is that of a vertex in the graph (but it could be that of a slow variable or a fast variable that is not represented in the graph) uncoupling is essential to keep the dynamics linear

21 but what about the other variables? the answer depends on the problem and the context in some cases, a property of the system is being calculated, like rate of product formation in enzyme kinetics or fractional saturation in protein allostery these properties depend on the vertices in the graph and the elimination formula then describes the properties in terms of the slow variables in other cases, the values of the slow variables must be calculated separately, for instance through conservation laws which are external to the graph, or by making further assumptions like the no depletion assumption in gene regulation (*) (*) Ahsendorf, Wong, Eils, Gunawardena, A graph-based framework for modelling gene regulation reveals history-depdendent complexity away from equilibrium, submitted, 2014.

22 reversible michaelis-menten k1[s] + k4[p] E + S k1 k2 ES k3 k4 E + P E ES k2 + k3 uncoupling condition nonlinear dynamics Laplacian (linear) dynamics

23 reversible michaelis-menten spanning trees k1[s] + k4[p] E k1[s] + k4[p] ES k2 + k3 elimination E ES k2 + k3 E ES

24 reversible michaelis-menten E + S k1 k2 ES k3 k4 E + P rational expression forward & reverse maximal rates forward & reverse Michaelis-Menten constants Athel Cornish-Bowden, Fundamentals of Enzyme Kinetics, 2nd edition, Portland Press, 2001

25 enzyme kinetics elimination rational expression Gulbinsky & Cleland, Kinetic studies of Escherichia coli galactokinase, Biochemistry 7: King, & Altman, A schematic method of deriving the rate laws for enzyme catalyzed reactions, J Phys Chem 60:

26 realistic enzymology Michaelis-Menten mechanism strongly irreversible (product does not rebind) E + S ES E + P E + S ES E + P reversible irreversible but not strongly (no conversion of P to S) E + S ES EP E + P enzyme mechanisms can be even more complicated with multiple intermediates and redundant pathways Xu, Gunawardena, J Theor Biol 311:

27 realistic enzymology for PTM systems grammar for expressing enzyme reaction mechanisms E + S0 Yi Yj Yk Ym E + S1 suitable for forward and reverse small-molecule modifications the linear framework shows that any reaction mechanism expressed in the grammar can be specified at steady state by 4 aggregated parameters total generalised catalytic efficiencies for S0 total generalised MichaelisMenten constants for S0 for S1 for S1 this allows us to prove results about PTM systems which hold irrespective of the mechanisms of the enzymes we can rise above (some) of the complexity Dasgupta, Croll, Owen, Vander Heiden, Locasale, Alon, Cantley, Gunawardena, J Biol Chem 289:

28 complexity reduction for PTM systems system with a single substrate S with multiple sites being modified in different ways by small-molecule modifications N astronomically large multiple forward and reverse modifying enzymes k relatively very small each enzyme using any reaction mechanism expressed in the grammar the linear framework gives the steady-state behaviour as the solution to k algebraic conservation equations for the free enzyme concentrations rational expressions the steady-state complexity of a PTM system depends on the number of enzymes (k), not on the number of modification states (N) we can rise above all the complexity Thomson & Gunawardena, Nature 460: ; J Theor Biol 261:

29 protein allostery enzymes lactate dehydrogenase ligand (F) relaxed tense elimination rational expression Monod, Wyman & Changeux, J Mol Biol 12: ; Koshland, Nemethy & Filmer, Biochem 5: ; Changeux & Edelstein, Science 308:

30 protein allostery receptors G-protein coupled receptor (GPCR) ion channels elimination elimination rational expressions Samama, Cotecchia, Cost, Lefkowitz, A mutation induced activation state of the beta2adrenergic receptor, J Biol Chem 268: Edelstein, Schaad, Henry, Bertrand, Changeux, A kinetic mechanism for nicotinic acetylcholine receptors based on multiple allosteric transitions, Biol Cybern 75:

31 gene expression (at thermodynamic equilibrium) Ackers, Johnson & Shea, PNAS 79: Bintu et al, Curr Opin Gen Dev 15: elimination Setty, Mayo, Surette & Alon, PNAS 100: rational expressions

32 gene regulation (away from equilibrium) Pho4(SA1-4PA6)-YFP UASp1 UASp2 TATA CFP -3 normalised CFP ([Pho5]) partial PHO5 promoter elimination normalised YFP ([Pho4]) GENE REGULATION FUNCTION rational expression Kim & O'Shea, A quantitative model of transcription factor-activated gene expression, Nature Struct Mol Biol 15: ; Ahsendorf, Wong, Eils, Gunawardena, A graph-based framework for modelling gene regulation reveals history-depdendent complexity away from equilibrium, submitted, 2014.