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18. Lecture WS 2006/07Bioinformatics III1 V18 Stochastic simulations of cellular signalling Traditional computational approach to chemical/biochemical kinetics: (a) start with a set of coupled ODEs (reaction rate equations) that describe the time-dependent concentration of chemical species, (b) use some integrator to calculate the concentrations as a function of time given the rate constants and a set of initial concentrations. Successful applications : studies of yeast cell cycle, metabolic engineering, whole-cell scale models of metabolic pathways (E-cell),... Major problem: cellular processes occur in very small volumes and frequently involve very small number of molecules. E.g. in gene expression processes a few TF molecules may interact with a single gene regulatory region. E.coli cells contain on average only 10 molecules of Lac repressor. Slide 2 18. Lecture WS 2006/07Bioinformatics III2 Include stochastic effects (Consequence1) modeling of reactions as continuous fluxes of matter is no longer correct. (Consequence2) Significant stochastic fluctuations occur. To study the stochastic effects in biochemical reactions stochastic formulations of chemical kinetics and Monte Carlo computer simulations have been used. Daniel Gillespie (J Comput Phys 22, 403 (1976); J Chem Phys 81, 2340 (1977)) introduced the exact Dynamic Monte Carlo (DMC) method that connects the traditional chemical kinetics and stochastic approaches. Assuming that the system is well mixed, the rate constants appearing in these two methods are related. Slide 3 18. Lecture WS 2006/07Bioinformatics III3 Dynamic Monte Carlo In the usual implementation of DMC for kinetic simulations, each reaction is considered as an event and each event has an associated probability of occurring. The probability P(E i ) that a certain chemical reaction E i takes place in a given time interval t is proportional to an effective rate constant k and to the number of chemical species that can take part in that event. E.g. the probability of the first-order reaction X Y + Z would be k 1 N x with N x :number of species X, and k 1 : rate constant of the reaction Similarly, the probability of the reverse second-order reaction Y + Z X would be k 2 N Y N Z. Resat et al., J.Phys.Chem. B 105, 11026 (2001) Slide 4 18. Lecture WS 2006/07Bioinformatics III4 Dynamic Monte Carlo As the method is a probabilistic approach based on events, reactions included in the DMC simulations do not have to be solely chemical reactions. Any process that can be associated with a probability can be included as an event in the DMC simulations. E.g. a substrate attaching to a solid surface can initiate a series of chemical reactions. One can split the modelling into the physical events of substrate arrival, of attaching the substrate, followed by the chemical reaction steps. Resat et al., J.Phys.Chem. B 105, 11026 (2001) Slide 5 18. Lecture WS 2006/07Bioinformatics III5 Basic outline of the direct method of Gillespie (Step i) generate a list of the components/species and define the initial distribution at time t = 0. (Step ii) generate a list of possible events E i (chemical reactions as well as physical processes). (Step iii) using the current component/species distribution, prepare a probability table P(E i ) of all the events that can take place. Compute the total probability P(E i ) : probability of event E i. (Step iv) Pick two random numbers r 1 and r 2 [0...1] to decide which event E will occur next and the amount of time by which E occurs later since the most recent event. Resat et al., J.Phys.Chem. B 105, 11026 (2001) Slide 6 18. Lecture WS 2006/07Bioinformatics III6 Basic outline of the direct method of Gillespie Using the random number r 1 and the probability table, the event E is determined by finding the event that satisfies the relation Resat et al., J.Phys.Chem. B 105, 11026 (2001) The second random number r 2 is used to obtain the amount of time between the reactions As the total probability of the events changes in time, the time step between occurring steps varies. Steps (iii) and (iv) are repeated at each step of the simulation. The necessary number of runs depends on the inherent noise of the system and on the desired statistical accuracy. Slide 7 18. Lecture WS 2006/07Bioinformatics III7 Weighted Sampling In the commonly used MC algorithm, the Markov chain is generated using transition probabilities (i j) that are based on the physical probability distribution: Resat et al., J.Phys.Chem. B 105, 11026 (2001) The ensemble average of any physical quantity is obtained by taking the arithmetic average of all the n simulation runs. The individual averages i could e.g. be time-averages over the simulation run. This choice disfavors the transitions with low probabilities. If the system characteristics depend on the events that happen less frequently, then the common implementation of MC requires extremely lengthy simulations to acquire enough statistical sampling. Slide 8 18. Lecture WS 2006/07Bioinformatics III8 Weighted Sampling This statistical sampling problem can be avoided if the probability distribution is multiplied with a weight function that adjusts the sampling probability distribution such that the low probability parts of the sampling space are visited more often. In the case of weighted sampling, the Markov chain is generated by using the modified probability distribution function Resat et al., J.Phys.Chem. B 105, 11026 (2001) where Y is the biasing weight function. Since the probability of the transition i j is weighted with Y(i j), calculation of the ensemble average of a physical quantity is obtained by computing the average of / Y. Division of by Y effectively corrects for the bias introduced in the sampling probability distribution. Slide 9 18. Lecture WS 2006/07Bioinformatics III9 Probability-Weighted DMC Probability-weighted DMC incorporates weighted sampling into DMC. Steps (iii) and (iv) of the DMC algorithm are replaced by (Step iii) Using the current component/species distribution, prepare a probability table of all the events E i that can take place, (Step iv) define the weight factor scale and compute the inverse probability weight table Resat et al., J.Phys.Chem. B 105, 11026 (2001) for all events. Note that the stochastic simulations mentioned here use discrete numbers of molecules, i.e. the species are produced and consumed as whole integer units. Therefore, the weight table w(E ) must contain only integer values. Slide 10 18. Lecture WS 2006/07Bioinformatics III10 Probability-Weighted DMC (Step v) Prepare the weighted probability table Resat et al., J.Phys.Chem. B 105, 11026 (2001) (Step vi) Compute the total probability by summing the weighted probabilities of all individual events (Step vii) Pick two random numbers r 1,r 2 [0...1]. Determine which event E occurs next as before using r 1. (Step viii) Propagate the time as before using r 2. The speed-up achieved by the PW-DMC algorithm stems from the fact that the reactions with large probabilities are allowed to occur in bundles. Slide 11 18. Lecture WS 2006/07Bioinformatics III11 Comparison of DMC and PW-DMC DMC is essentially a method to solve the master equation that rules how the probabilities of the configurations are related to each other Resat et al., J.Phys.Chem. B 105, 11026 (2001) W : transition probability of going from configuration to P : probability of configuration . Using the master equation, the statistical average X of the rate of change of the property X can be expressed as: In PW-DMC, this relation is rearranged using the weight factor w as PW-DMC leaves the ensemble averages unchanged. However, the fluctuations increase with w. Slide 12 18. Lecture WS 2006/07Bioinformatics III12 Integrated Model of Epidermal Growth Factor Receptor Trafficking and Signal Transduction The EGF receptor can be activated by the binding of any one of a number of different ligands. Each ligand stimulates a somewhat different spectrum of biological responses. The effect of different ligands on EGFR activity is quite similar at a biochemical level the mechanisms responsible for their differential effect on cellular responses are unkown. After binding of any of its ligands, EGFR is rapidly internalized by endocytosis. Resat et al. Biophys Journal 85, 730 (2003) Slide 13 18. Lecture WS 2006/07Bioinformatics III13 Computational modelling of EGF receptor system (1)trafficking and ligand-induced endocytosis (2)signaling through Ras or MAP kinases This work combines both aspects into a single model. Most approaches to building computational kinetic models have severe drawbacks when representing spatially heterogenous processes on a cellular scale. Review: In the traditional approach, we - formulate set of coupled ODEs (reaction rate equations) for the time-dependent concentration of chemical species - use integrator to propagate the concentrations as a function of time given the rate constants and a set of initial concentrations. Resat et al. Biophys Journal 85, 730 (2003) Slide 14 18. Lecture WS 2006/07Bioinformatics III14 Multiple time scale problem In Dynamic Monte Carlo, reactions are considered events that occur with certain probabilities over set intervals of time. The event probabilities depend on the rate constant of the reaction and on the number of molecules participating in the reaction. In many interesting natural problems, the time scales of the events are spread over a large spectrum. Therefore it is very inefficient to treat all processes at the time scale of the fastest individual reaction. In the EGFR signaling network, - receptor phosphorylation after ligand binding occurs almost instantaneously - vesicle formation or sorting to lysosomes requires many minutes. Resat et al. Biophys Journal 85, 730 (2003) Slide 15 18. Lecture WS 2006/07Bioinformatics III15 Solution to multiple time scale problem Computing millions and billions non-correlated random numbers can become a time-consuming process. Resat et al. (2001) introduced Probability-Weighted DMC to speed-up the simulation by factor 20 100. Different processes are only tested at variant times depending on their probabilities = very unlikely processes compute MC decision very infrequently. Resat et al. Biophys Journal 85, 730 (2003) Slide 16 18. Lecture WS 2006/07Bioinformatics III16 Signal transduction model of EGF receptor signaling pathway Resat et al. Biophys Journal 85, 730 (2003) Slide 17 18. Lecture WS 2006/07Bioinformatics III17 Species in the EGF receptor signaling model Resat et al. Biophys Journal 85, 730 (2003) Slide 18 18. Lecture WS 2006/07Bioinformatics III18 Receptor and ligand group definitions Resat et al. Biophys Journal 85, 730 (2003) Slide 19 18. Lecture WS 2006/07Bioinformatics III19 Early endosome inclusion coefficients Resat et al. Biophys Journal 85, 730 (2003) These are adjusted to yield the experimentally determined rates of ligand-free and ligand-bound receptor internalization. Slide 20 18. Lecture WS 2006/07Bioinformatics III20 Time course of phosphorylated EGF receptors (a) Total number of phosphorylated EGF receptors in the cell. Curves represent the number of activated receptors when the cell is stimulated with different ligand doses at the beginning. The y axis represents the number of receptors in thousands. (b ) Ratio of the number of phosphorylated receptors that are internalized to that of the phosphorylated surface receptors. (c) Ratio of the number of internalized receptors to the number of surface receptors. Curves are colored as: [L] = 0.2 (magenta), 1 (blue), 2 (green), and 20 (red) nM. Resat et al. Biophys Journal 85, 730 (2003) Slide 21 18. Lecture WS 2006/07Bioinformatics III21 Distribution of the receptors among cellular compartments Resat et al. Biophys Journal 85, 730 (2003) Slide 22 18. Lecture WS 2006/07Bioinformatics III22 Stimulation of EGFR signaling pathway by different ligands Comparison of the results when the EGFR signaling pathway is stimulated with its ligands EGF (red) and TGF- (green). (a ) Total number of receptors in the cell as a function of time after 20 nM ligand is added to the system. Red diamond (EGF) and green square (TGF- ) points show the experimental results. (b) Distribution of the receptors between intravesicular compartments and the cell membrane. (c) Distribution of the phosphorylated receptors between intravesicular compartments and the cell membrane. In the figures, y axes represent the number of receptors in thousands. Resat et al. Biophys Journal 85, 730 (2003) Slide 23 18. Lecture WS 2006/07Bioinformatics III23 Ratio of internal/surface receptors The ratio of the In/Sur ratios when the EGFR signaling pathway is stimulated with its ligands EGF and TGF- at 20 nM ligand concentration. Comparison of computational (solid lines) and experimental (points) results. Ratio of the ratios for the phosphorylated (i.e., activated) (blue), and total (phosphorylated + unphosphorylated) number (magenta) of receptors. Resat et al. Biophys Journal 85, 730 (2003) Slide 24 18. Lecture WS 2006/07Bioinformatics III24 Summary Large-scale simulations of the kinetics of biological signaling networks are becoming feasible. Here, the model consisted of hundreds of distinct compartments and ca. 13.000 reactions/events that occur on a wide spatial-temporal range. The exact Dynamic Monte Carlo algorithm of Gillespie (1976/1977) was a breakthrough for simulations of stochastic systems. Problem: simulations can become very time-consuming. In particular if the processes occur on different time scales. Methods like the probability-weighted DMC are promising tools for studying complex cellular systems using molecular quanta. Many other variants of DMC have and are being development. Slide 25 18. Lecture WS 2006/07Bioinformatics III25 Bacterial Photosynthesis 101 Photons Light Harvesting Complexes light energy electronic excitation Reaction Center e H + pairs ATPase chemical energy cytochrome bc 1 complex H + gradient; transmembrane potential ubiquinon cytochrome c 2 electron carriers outside inside Slide 26 18. Lecture WS 2006/07Bioinformatics III26 Modelling as metabolic network Chemical reactions involved: Slide 27 18. Lecture WS 2006/07Bioinformatics III27 Photosynthesis cycle view light energy electronic excitation e H + pairs chemical energy H + gradient, transmembrane voltage outside inside The conversion chain: stoichiometries must match turnovers! electrons 2 cycles: protons Slide 28 18. Lecture WS 2006/07Bioinformatics III28 LH1 / LH2 / RC a la textbook Collecting photons Hu et al, 1998 B800, B850, Car. LH2: 8 dimers LH1: 16 dimers downhill transport of excitons LH2 LH1 RC Slide 29 18. Lecture WS 2006/07Bioinformatics III29 The Cytochrome bc 1 complex the "proton pump" X-ray structures known Berry, etal, 2004 always forms a dimer Q-cycle: 2H + per 1e Slide 30 18. Lecture WS 2006/07Bioinformatics III30 The F o F 1 -ATP synthase I at the end of the chain: producing ATP from the H+ gradient Capaldi, Aggeler, 2002 per turn: 1014 H + 3 ATP 1 ATP 4 H + Slide 31 18. Lecture WS 2006/07Bioinformatics III31 The electron carriers Cytochrome c: electrons from bc 1 to RC heme in a hydrophilic protein shell 3.3 nm diameter Ubiquinon UQ10: carries electronproton pairs from RC to bc 1 hydrophobic tail long (2.4 nm) isoprenoid tail taken from Stryer Slide 32 18. Lecture WS 2006/07Bioinformatics III32 Tubular membranes photosynthetic vesicles where are the bc 1 complexes and the ATPase? Jungas et al., 1999 200 nm LH1 RC bc1? * 50 nm 100 nm Bahatyrova et al., 2004 no bc 1 found! Slide 33 18. Lecture WS 2006/07Bioinformatics III33 Chromatophore vesicle: typical form in Rh. sphaeroides Lipid vesicles 3060 nm diameter H + and cyt c inside Vesicles are really small! average chromatophore vesicle, 45 nm : surface 6300 nm Slide 34 18. Lecture WS 2006/07Bioinformatics III34 Photon capture rate of LHCs + Bchl extinction coeff. normalization ( Bchl = 2.3 2 ) relative absorption spectrum of LH1/RC and LH2 sun's spectrum at ground (total: 1 kW/m) multiply capture rate: 0.1 s kW Bchl typical growth condition: 18 W/m LH1: 16 * 3 Bchl 14 /s LH2: 10 * 3 Bchl 10 /s Cogdell etal, 2003 Feniouk et al, 2002 Franke, Amesz, 1995 Wavelength [nm] dE/d [arb.] Gerthsen, 1985 Slide 35 18. Lecture WS 2006/07Bioinformatics III35 LH1 / LH2 / RC native Siebert etal, 2004 electron micrograph and density map 125 * 195 , = 106 Area per: per vesicle (45 nm) LH1 monomer (hexagonal) 146 nm LH1 dimer234 nm LH2 monomer37 nm LH1 2 + 6 LH2456 nm11 Chromatophore vesicle, 45 nm : surface 6300 nm Slide 36 18. Lecture WS 2006/07Bioinformatics III36 Photon processing rate at the RC Which process limits the RCs turnover? Unbinding of the quinol 25 ms Milano et al. 2003 + binding, charge transfer 50 ms per quinol (estimate) with 2e - H + pairs per quinol 4050 /s per RC 22 QH 2 /s 1 RC can serve 1 LH1 + 3 LH2 = 44 /s LH1 2 + 6 LH2 456 nm 11 LH1 dimers including 22 RCs on one vesicle 480 Q/s can be loaded @ 18 W/m per vesicle Slide 37 18. Lecture WS 2006/07Bioinformatics III37 The F 1 F 0 -ATP synthase "mushroom like structures observed in AFM images" ATPase is "visible" 1 ATPase per vesicle Feniouk et al, 2002 Grber et al, 1991, 1999 limited throughput of the ATPase "Arrhenius" "binding" per turn: 1014 H + per 3 ATP 1 ATP 4 H + ATPase fromATP/sH + /s chloroblasts