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.

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.

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(Ei) that a certain chemical reaction Ei 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 + Zwould be k1Nx with Nx :number of species X, and k1 : rate constant of the reaction

Similarly, the probability of the reverse second-order reactionY + Z Xwould be k2NYNZ.Resat et al., J.Phys.Chem. B 105, 11026 (2001)

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)

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 Ei (chemical reactions as well as physical processes).

(Step iii) using the current component/species distribution, prepare a probability table P(Ei) of all the events that can take place.Compute the total probability

P(Ei) : probability of event Ei .

(Step iv) Pick two random numbers r1 and r2 [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)

Basic outline of the direct method of GillespieUsing the random number r1 and the probability table,the event E is determined by finding the event that satisfies the relationResat et al., J.Phys.Chem. B 105, 11026 (2001)The second random number r2 is used to obtain the amount of time between the reactionsAs 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.

Weighted SamplingIn 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.

Weighted SamplingThis 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 functionResat 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.

Probability-Weighted DMCProbability-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 Ei that can take place,

(Step iv) define the weight factor scale and compute the inverse probability weight tableResat 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.

Probability-Weighted DMC(Step v) Prepare the weighted probability tableResat 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 r1,r2 [0...1].

Determine which event E occurs next as before using r1.

(Step viii) Propagate the time as before using r2.

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.

Comparison of DMC and PW-DMCDMC is essentially a method to solve the master equation that rules how the probabilities of the configurations are related to each otherResat 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 asPW-DMC leaves the ensemble averages unchanged.However, the fluctuations increase with w.

Integrated Model of Epidermal Growth Factor Receptor Trafficking and Signal TransductionThe 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)

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)

Multiple time scale problemIn 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)

Solution to multiple time scale problemComputing 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)

Signal transduction model of EGF receptor signaling pathwayResat et al. Biophys Journal 85, 730 (2003)

Species in the EGF receptor signaling modelResat et al. Biophys Journal 85, 730 (2003)

Receptor and ligand group definitionsResat et al. Biophys Journal 85, 730 (2003)

Early endosome inclusion coefficientsResat et al. Biophys Journal 85, 730 (2003)These are adjusted to yield the experimentally determined rates ofligand-free and ligand-bound receptor internalization.

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)

Distribution of the receptors among cellular compartmentsResat et al. Biophys Journal 85, 730 (2003)

Stimulation of EGFR signaling pathway by different ligandsComparison 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)

Ratio of internal/surface receptorsThe 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)

SummaryLarge-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.

Bacterial Photosynthesis 101PhotonsLight Harvesting Complexeslight energyelectronic excitationReaction CentereH+pairsATPasechemical energycytochrome bc1 complexH+ gradient; transmembrane potentialubiquinoncytochrome c2electron carriersoutsideinside

Modelling as metabolic networkChemical reactions involved:

Photosynthesis cycle viewlight energyelectronic excitationeH+pairschemical energyH+ gradient,transmembrane voltageoutsideinsideThe conversion chain: stoichiometries must match turnovers!

LH1 / LH2 / RC a la textbookCollecting photonsHu et al, 1998

The Cytochrome bc1 complexthe "proton pump"

The FoF1-ATP synthase Iat the end of the chain: producing ATP from the H+ gradient

The electron carriersCytochrome c: electrons from bc1 to RC heme in a hydrophilic protein shell 3.3 nm diameterUbiquinon UQ10: carries electronproton pairs from RC to bc1 hydrophobic tail long (2.4 nm) isoprenoid tailtaken from Stryer

Tubular membranes photosynthetic vesicleswhere are the bc1 complexes and the ATPase?Jungas et al., 1999

Chromatophore vesicle: typical form in Rh. sphaeroidesLipid vesicles3060 nm diameterH+ and cyt c insideVesicles are really small!average chromatophore vesicle, 45 nm :surface 6300 nm

Photon capture rate of LHCs+ Bchl extinction coeff.normalization (Bchl = 2.3 2)relative absorption spectrumof LH1/RC and LH2sun's spectrum at ground(total: 1 kW/m)multiplytypical growth condition: 18 W/mLH1: 16 * 3 Bchl 14 /sLH2: 10 * 3 Bchl 10 /sCogdell etal, 2003Feniouk et al, 2002Franke, Amesz, 1995Gerthsen, 1985

LH1 / LH2 / RC nativeSiebert etal, 2004electron micrographand density map125 * 195 , = 106Chromatophore vesicle, 45 nm :surface 6300 nm

Area per:per vesicle (45 nm)LH1 monomer(hexagonal)146 nmLH1 dimer234 nmLH2 monomer37 nmLH12 + 6 LH2456 nm11

Photon processing rate at the RC Which process limits the RCs turnover?Unbinding of the quinol 25 msMilano et al. 2003

+ binding, charge transfer 50 ms per quinol (estimate)

with 2e- H+ pairs per quinol 4050 /s per RC 22 QH2/s1 RC can serve 1 LH1+ 3 LH2= 44 /sLH12 + 6 LH2 456 nm 11 LH1 dimers including 22 RCs on one vesicle 480 Q/s can be loaded @ 18 W/m per vesicle

The F1F0-ATP synthase"mushroom like structures observed in AFM images" ATPase is "visible"per turn: 1014 H+ per 3 ATP

How much bc1?measured enzymatic activity per bc1 dimer: 84 cyt c are reduced / s400 H+ / s per ATPase 1 dimer can process 42 QH / s 1 dimer "pumps" 168 H+ / s 11 bc1 dimers per vesiclecan be loaded by RCs proton generation by 2.4 bc1 dimers per vesicleenough to drive 1 ATPasex5 !!!inoutsafety first!number of bc1 complexesshould be limited by howmany protons can bepumped out by ATPaseXiao et al. 2000

bc1 Placement Diffusional limits?Roundtrip timesmaximal capacity of the carriers:T = TRC + Tbc1 + TDff Cytochrome c:TRC 1 msTbc1 12 msTDiff 3 sTround-trip = 13 ms 3 cyt c per vesiclesufficient to carry e-savailable: 22 cyt c per vesicle Quinol:TRC 50 msTbc1 23 msTDiff 1 msTround-trip = 75 ms 7 Q per vesicle sufficient to carry e-s.available: 100 Q per vesicle poses no constraints for the position of bc1

Proposed setup of a chromatophore vesicleblue: small LH2 rings (blue)

blue/red: Z-shaped LH1/RC dimers form a linear array around the equator of the vesicle, determining the vesicles diameter by their intrinsic curvature. At the polesgreen/red: the ATPase light blue: the bc1 complexes

Increased proton density close to the ATPase.favors close placing ATPase and bc1 complexes.yellow arrows: diffusion of the protons out of the vesicle via the ATPase and to the RCs and bc1s.Geyer, Helms, Biophys. J. (2006) 91, 921

reconstituted LH1 dimers in planar lipid membranesUpper panel: drawn after the AFM images of Scheuring et al (4) of LH1 dimers reconstituted into planar lipid membranes. The Z-shaped dimers are seen from their cytoplasmic side. The average height of the membrane is indicated in the upper panel by the grey level of the background. Scheuring et al measured the distances between the minima of the AFM height scan to be 38.5 nm and the distance be tween the lowest and the highest parts to be 3.8 nm.Lower panel: interpretation how alternatingly oriented LH1 dimers may explain the observed height scan indicated by the dashed lines. These values are nicely reproduced by the proposed arrangement of the LH1 dimers, when one assumes that they are stiff enough to retain the bending angle of 26 that they would have on a spherical vesicle of 45 nm diameter and taking into account the length of a single LH1 dimer of about 19.5 nm.

Photosynthesis: textbook view

Viewing the photosynthetic apparatus as a conversion chainThick arrows : path through which the photon energy is converted into chemical energy stored in ATP via the intermediate stages (rounded rectangles).

Each conversion step takes place in parallely working proteins. Their number N times the conversion rate of a single protein R determines the total throughput of this step.

: incoming photons collected in the LHCsE : excitons in the LHCs and in the RCeH+ electronproton pairs stored on the quinolse for the electrons on the cytochrome c2pH : transmembrane proton gradientH+ : protons outside of the vesicle (broken outine of the respective reservoir).

Modelling of internal processes at reaction centerAll individual reactions with their individual rates k together determine the overall conversion rate RRC of a single RC. Thick arrows : flow of the energy from the excitons through the cyclic charge state changes of the special pair Bchl (P) of the RC. Rounded rectangles : reservoirs

Parameters

Stochastic dynamics simulations II Production runsfixed set of parameters

integrate rate equations with:

- Gillespie algorithm (associations)

- Timer algorithm (reactions); 1 random number determines when reaction occurs

example: binding and e- transfer at reaction center

Stochastic simulations of a complete vesicleModel vesicle:12 LH1/RC-monomers1-6 bc1 complexes1 ATPase

120 quinones20 cytochrome c2

integration time step: 10 s

simulating 1 minute real time requires 1.5 minute on one opteron 2.4 GHz proc

simulate increase of light intensityduring 1 minute,light intensity was slowly increased from 0 to 10 W/m2(quasi steady state)

there are two regimes- one limited by available light- one limited by bc1 throughputlow light intensity:linear increase of ATP production with light intensityhigh light intensity:saturation is reached the later the higher the number of bc1 complexesGeyer, Lauck, Helms, J. Biotech (to be accepted)

oxidation state of cytochrome c2 poollow light intensity:all 20 cytochrome c2are reduced by bc1high light intensityRCs are faster than bc1,c2s wait for electrons

oxidation state of cytochrome c2 poolmore bc1 complexescan load more cytochrome c2s

total number of produced ATPlow light intensity: any interruption stops ATP production

high light intensity: interruptions are buffered up to 0.3 s durationblue line:illumination

c2 pool acts as bufferAt high light intensity, c2 pool is mainly oxidized.

If light is turned off, bc1 can continue to work (load c2s, pump protons, let ATPase produce ATP) until c2 pool is fully reduced.

SummaryChromatophore vesicles appear an ideal model system to interface molecular simulations and ODE models.

Even though some parameters are still not known experimentally, the available knowledge allows to construct detailed kinetic and spatial models.

Such systems allow to test how much knowledge is required about a particular system.

Predictions need to be tested by new experiments!Such tests are essential before one can call a model correct.

T_{c_2} = 2\, \frac{(2R_i)^2}{6\, D_0}T_Q = 2\, \frac{(\pi\,R_m)^2}{4\, D_Q}