Reverse engineering of gene regulatory networks using information theory models has received much attention due to its simplicity, low computational cost, and capability of inferring large networks. One of the major problems with information theory models is to determine the threshold which defines the regulatory relationships between genes. The minimum description length (MDL) principle has been implemented to overcome this problem. The description length of the MDL principle is the sum of model length and data encoding length. A user-specified fine tuning parameter is used as control mechanism between model and data encoding, but it is difficult to find the optimal parameter. In this work, we proposed a new inference algorithm which incorporated mutual information (MI), conditional mutual information (CMI) and predictive minimum description length (PMDL) principle to infer gene regulatory networks from DNA microarray data. In this algorithm, the information theoretic quantities MI and CMI determine the regulatory relationships between genes and the PMDL principle method attempts to determine the best MI threshold without the need of a user-specified fine tuning parameter.Saccharomyces cerevisiae. The benchmark quantities precision and recall were used as performance measures. The results show that the proposed algorithm produced less false edges and significantly improved the precision, as compared to the existing algorithm. For further analysis the performance of the algorithms was observed over different sizes of data.The performance of the proposed algorithm was evaluated using both synthetic time series data sets and a biological time series data set for the yeast We have proposed a new algorithm that implements the PMDL principle for inferring gene regulatory networks from time series DNA microarray data that eliminates the need of a fine tuning parameter. The evaluation results obtained from both synthetic and actual biological data sets show that the PMDL principle is effective in determining the MI threshold and the developed algorithm improves precision of gene regulatory network inference. Based on the sensitivity analysis of all tested cases, an optimal CMI threshold value has been identified. Finally it was observed that the performance of the algorithms saturates at a certain threshold of data size. Bayesian Networks .R) and precision (P) are used to evaluate the performance of inference algorithms. While different definitions exist [R is herein defined as eC/(eC+eM) and P as eC/(eC+eF), where eC denotes the edges that exist in both the true and the inferred network, eM are the edges that exist in the true network but not in the inferred network and eF are the edges that do not exist in the true network but do exist in the inferred network.The proposed algorithm is compared with  on synthns exist , R is heP and R are calculated. The algorithms are run for 20, 30, 40 and 50 numbers of genes. The P vs. R curves for each of these networks with different threshold values are given in Figure For a specific size of the network, both the algorithms are run for different threshold values 30 times each and the average of Zhao et al.  reportedP of the proposed algorithm is higher but R is lower in most of the cases as compared to network MDL. The number of false negatives is fewer in the proposed algorithm and as most biologists are interested in true positives, our proposed algorithm is preferred over the network MDL.In Figure The time series DNA microarray data from  was usedAs mentioned earlier, pre-processing plays an important part in reverse engineering process. As there were some missing values in the data we pre-processing the data as in . InitialThe true biological network used for comparison purposes was derived from the yeast cell cycle pathway -25. A toHere we report the performance of our scheme based on different values of the user specified threshold over synthetic networks. For threshold values of 0.15 and 0.2, a high precision (over 90% in most cases) was observed but the recall for these thresholds was low (from 25% to 30%) as compared to a threshold value of 0.1 which had a fair recall (over 47%) and good precision (63% to 79%) performance.Figure The performance of the algorithm depends on three factors. The number of genes, the number of time points and most importantly the number of parents inferred for each gene by the algorithm. To see what role these factors play we looked into the time and space complexities of the algorithm.n2m times, where n is number of genes and m is the number of time points; from line 5 to line 18 the algorithm iterates n4 times; lines 15 and 16 of the algorithm iterates n3m times. Finally, from lines 20 to 31 the algorithm iterates n3 times. Thus, the time complexity of the algorithm is Θ(n4 + n3m).Step 4 of the algorithm iterates From the time complexity it can be seen that if the number of genes is larger than the number of time points then the run time depends more on the number of genes. And if the number of time points is larger than the number of genes then the run time depends more on the number of time points.n parents then the conditional probability tables take 2n units of space. Thus, the amount of memory needed by the algorithm depends on the number of parents inferred by the network. As the space complexity grows exponentially based on the number of parents it is possible that the algorithm may run out of memory for a data set with as few as 50 genes but run for as little as 5 minutes for a data set with several hundred genes. There are 2 ways to overcome this limitation: When it comes to space complexity the conditional probability tables play a major role. If a gene has 1. Restrict the number of parents and2. Take the next smallest description length, instead of using the smallest one.The first approach guarantees results when the number of parents is restricted to small values but this may lower the accuracy of the result. The second approach may take more time to run but as we are not restricting the number of parents the accuracy of the algorithm is not affected. We plan to perform some bench marking studies on the above two approaches to see which one works better.A network and a data set with 75 time points were generated. Each of the algorithms was run 13 times starting with the first 15 time points. An increment of 5 time points was made for every subsequent run. For every run the values of precision and recall were computed. In Network MDL the free parameter was set to 0.2 and in PMDL algorithm the conditional mutual information threshold was set to 0.1. The plots for precision and recall are as shown in Figure For the PMDL algorithm it is observed from Figure For further analysis we considered the recall/precision ratio Figure . SaturatWe have proposed a new gene regulatory inference algorithm that implements the PMDL principle. The simulation results show that the PMDL principle is fair in determining the MI threshold. A problem with the proposed algorithm is determining the CMI threshold. We have tested the sensitivity of the threshold and based on the performance of our scheme we identified that the value of 0.1 is optimal for most synthetic networks. This was also true in the case of reverse engineering of gene regulatory networks from biological time series DNA microarray data. In synthetic network simulations the proposed algorithm produced fewer false edges compared to ; howeverG represents a network where V denotes a set of genes and E denotes a set of regulatory relationships between genes. If gene x shares a regulatory relationship with gene y, then there exists an edge between x and y (x → y). Genes can have more than one regulator. The notation P(x) is used to represent a set of genes that share regulatory relationships with gene x. For example, if gene  shares a regulatory relationship with y and z then P(x) = {y, z}. Also every gene is associated with a function xPf(x) which denotes the expression value of gene x determined by the values of genes in P(x). The network formulation is similar to the one used in . A graphq levels and a gene x has n predecessors then the look up table has nq rows and q columns and every entry in the table corresponds to a conditional probability.The gene expression is affected by many environmental factors. Since it is not possible to incorporate all factors the regulatory functions are assumed to be probabilistic. Also, the gene expression values are assumed discrete-valued and the probabilistic regulation functions are represented as look-up tables. If the expression levels are quantized to x which shares regulatory relationship with two other genes y,z and the data is quantized to 2 levels, the look up table is as in Table y and z are lowly expressed then the probability that x is also lowly expressed is 0.6.Say we have a gene Entropy: Entropy (H) is the measure of average uncertainty in a random variable. Entropy of a random variable X with probability mass function p(x) is defined [ defined  byMutual information: MI measures the amount of information that can be obtained about one random variable by observing another one. Since MI by itself does not contain directional information, using ad hoc time delay has been proposed in the past to overcome this issue. The gene system is assumed to be event driven, i.e. all the regulations are performed step by step and in each step all regulations happen only once. Therefore, the latency parameter is set by default to a unit step.MI is defined  asMI can also be defined in terms of entropies as I = H(X) + H(Y) − H	(3)As stated earlier this above quantity does not contain directional information and hence a time lag is introduced and the quantity after time lag is estimated as:I = H(tX) + H(Yt+1) − H	(4)Figure Conditional mutual information: High MI indicates that there may be a direct or indirect relationship between the genes. CMI is the reduction in the uncertainty of X due to knowledge of Y when Z is given [X and Y given Z is defined [is given . Time la defined  asCMI can also be expressed in terms of entropies as:I = H + H − H(Z) − H	(6)Again this quantity does not contain directional information. After introducing time lag the quantity is estimated as:I = H + H − H(tZ) − H	(7)Figure qA = {0, 1, …, q − 1} then, the probability mass function from m amples s1, …, ms is estimated asThe proposed algorithm deals with quantized data. In general, it is assumed that the q-level quantization admits the alphabet {.}(.) is the indicator function, defined asWhere, 1By substituting (9) in (1) entropies can be estimated and these entropy estimates can be substituted in (3) and (6) to obtain the MI and CMI estimates .The description length of the two-part MDL principle involves calculation of the model length and the data length. As the length can vary for various models, the method is in danger of being biased towards the length of the model . The NorWe chose to implement the PMDL principle as it suits time series data .  The coThe description length for a model in PMDL ,26 is gip(Xt+1|tX) is the conditional probability or density function. We calculate the description length as data length given in [Where given in .tX = T and the transition probability p(Xt+1|tX) can be derived as follows:A gene can take any value when transformed from one time point to another due to the probabilistic nature of the network. The network is associated with Markov chain which is used to model state transitions. These states are represented as n-gene expression vectors p can be obtained from the look-up table associated with the vertex ix and is assumed to be time invariant. It is estimated as follows:The probability Each state transition brings new information that is measured by the conditional entropy:   H(Xt+1|tX)= − log(p(Xt+1|tX))	(12)X1, …, mX) is given byThe total entropy for given m time-series sample points, (H(X1) is same for all models it is removed thus the description length is As Mn×n was evaluated using (4). A connectivity matrix Cn×n was maintained which had two entries: 0 and 1. An entry of 0 indicates that no regulatory relationship exists between genes, but an entry of 1 at Ci×j indicates that gene i regulates j. The algorithm is given in Figure Th) the connection is deleted.Given the time series data, the data was first pre-processed, which involved filling missing values and quantizing the data. Then the MI matrix The authors declare that they have no competing interests.VC, CZ and PG1 developed the algorithm and implemented the algorithm on synthetic and biological data sets. An in-depth analysis of results was also performed on the results. EP, PG2 and YD coordinated the study. All authors read and approved the final manuscript.


Rhodobacter sphaeroides as a model system, we show that a consistent kinetic model emerges when fitting the dynamics of a molecular stochastic simulation to a set of time dependent experiments even though about two thirds of the kinetic parameters in this system are not known from experiment. Those kinetic parameters that were previously known all came out in the expected range. The simulation model was built from independent protein units composed of elementary reactions processing single metabolites. This pools-and-proteins approach naturally compiles the wealth of available molecular biological data into a systemic model and can easily be extended to describe other systems by adding new protein or nucleic acid types. The automated parameter optimization, performed with an evolutionary algorithm, reveals the sensitivity of the model to the value of each parameter and the relative importances of the experiments used. Such an analysis identifies the crucial system parameters and guides the setup of new experiments that would add most knowledge for a systemic understanding of cellular compartments. The successful combination of the molecular model and the systemic parametrization presented here on the example of the simple machinery for bacterial photosynthesis shows that it is actually possible to combine molecular and systemic modeling. This framework can now straightforwardly be applied to other currently less well characterized but biologically more relevant systems.Metabolic processes in biological cells are commonly either characterized at the level of individual enzymes and metabolites or at the network level. Often these two paradigms are considered as mutually exclusive because concepts from neither side are suited to describe the complete range of scales. Additionally, when modeling metabolic or regulatory cellular systems, often a large fraction of the required kinetic parameters are unknown. This even applies to such simple and extensively studied systems like the photosynthetic apparatus of purple bacteria. Using the chromatophore vesicles of  Modern computational systems biology aims at an overall description of the components, interactions, regulatory circuits, and metabolic fluxes in biological cells Here, we show that this gap between molecular and systems biology can be successfully bridged by combining our previously presented pools-and-proteins approach Rhodobacter (Rb.) sphaeroides and compared the dynamic behavior of a molecular-stochastic model to a set of time-dependent experiments. The selected experiments were taken from a project which investigated the role of the PufX protein in cyclic electron transfer. They were conducted in the group of Oesterhelt and published in two consecutive papers Rb. sphaeroides, which can live on respiration, fermentation, or photosynthesis, is one of the best studied bacteria bc1 complex, and the FO-F1-ATP synthase (ATPase). It also contains the two electron carriers cytochrome c2 and ubiquinone (Q). The chromatophore vesicles have an average diameter of 45 to 60 nm only c oxidase from the respiratory chain, experimental studies of the photosynthetic apparatus typically poison these proteins by adding potassium cyanide, so that there is no interference with this competing metabolic pathway b site. The charge of the electron on the Q is compensated by a proton taken up from the cytoplasm. Loaded with two such electron-proton pairs, the resulting quinol (QH2) unbinds and diffuses inside the membrane to the cytochrome bc1 complex. There, the protons are released into the vesicle interior and the energy of the two electrons is used in the so called Q-cycle To demonstrate the power of such a bottom-up modeling approach combined with a systemic parameter determination, we used the simple and well understood photosynthetic apparatus of the purple bacterium c2 and ubiquinone by using experimentally determined stoichiometries and spatial constraints such as the surface area of the vesicle Rb. sphaeroides of 45 nm diameter contains about ten dimeric core complexes of two RCs and a Z-shaped LH1 each bc1 complexes could only be determined by us with a rather large uncertainty. Many other kinetic parameters were left unconstrained, too, by this steady state reconstruction. A systems biological steady state reconstruction based on elementary modes was recently presented by Klamt et al Although bacterial photosynthesis is usually considered well understood, the available descriptions are rather qualitative representations similar to bc1 complex was extended from its original implementation in order to include more biological detail required to extract quantities that were measured in the experiments. The main focus of this publication is placed on the systemic parametrization of the underlying molecular stochastic model. By fitting against experiments covering all these time regimes Rb. sphaeroides turned out to be an ideal system to develop and test our approach, which now allows to straightforwardly bridge between the molecular and the systemic realms of computational biology for other systems of more current interest, too.As light serves as the central in-flux “metabolite”, the photosynthetic apparatus can be probed very easily in a wide range of kinetic regimes. These range from the picosecond time-scale for electronic excitations in the LHCs and RCs probed by single flash experiments over association and dissociation reactions on the millisecond range up to quasi steady state conditions under constant illumination. We previously presented a first model how the encapsulated objects representing the individual proteins are built up from their microscopic reactions However, the model of the bacterial photosynthesic apparatus presented here is far from complete. It is focussed on the metabolic processes taking place on the millisecond to second timescales, whereas any regulatory adaptations to slowly changing environmental conditions are omitted. On the other hand, the very fast multi-step exciton and electron dynamics in the LHCs are described by a few low-detail effective reactions in the current iteration of the model. This simplified picture was sufficient because these processes were not resolved in the experiments. These two examples show that the model of bacterial photosynthesis as presented here serves well as a versatile scaffold or template that can be used to collect knowledge about the individual proteins and about how they interact to reproduce the rich spectrum of experimental observations. It nevertheless proofs that this concept can be used to bridge between the two currently not so well connected fields of molecular and systems biology.bc1 complexes affect the non-equilibrium dynamics of the chromatophore vesicles. This initial analysis is then followed by the main part of this work, the stochastic dynamics simulations combined with an evolutionary parameter optimization of the full vesicles.In this work we report on how we connected a molecular model of a biological system to its systemic treatment to bridge between these two fields. This two-sided approach is reflected in the following section, too, where we first explain the molecular modeling aspects required for the stochastic simulations, which serve as the starting point from the one side of the gap, and then how we performed the optimization by treating the complete model as one entity. Each of the tasks could be performed independently but by combining them we are able to connect the molecular and the systemic sides by exploiting the respective strenghts of both approaches. After the model setup we show how different numbers of cytochrome Rhodobacter sphaeroides presented in this work were performed with a refined version of the model introduced in The stochastic simulations of the chromatophore vesicles from c2 are two distinct species. As the chromatophore vesicles are so small that the diffusive transport through the vesicle is much faster than the respective times for association and dissociation In our model, the proteins are connected to each other via metabolite pools which are well-mixed subvolumes of the simulated system. Hereby, different charge or oxidation states of the same biological molecule are treated as different metabolites. For example, the oxidized and reduced forms of cytochrome c2 are reduced. Consequently, all c2s will be in the pool representing the reduced cytochrome 2c while the pool of oxidized 2c is empty. At the same time, about 90% of the quinones will be in the QH2 pool. Before the actual experiments were simulated by “exposing” the model vesicle to the respective illumination profiles, the simulations were “thermalized” for 20 to 100 milliseconds in the dark.The initial conditions of a simulation are set via the numbers of metabolites in the pools. In a dark-adapted vesicle all cytochrome t is determined. The probability Pon for an association reaction at a certain binding site of an individual protein during Δt depends on the concentration ρ of the metabolite in the pool and the association rate kon asτoff is determined following Gillespie koff and an exponentially distributed random number r:off a timer is initialized which then initiates the actual dissociation event. Note that such a waiting time algorithm cannot be used for the density dependent associations without extensive bookkeeping, because the metabolite concentrations may change after the timer was initialized due to other reactions that produce or consume the same metabolites.The state vector of the simulation is propagated by looping over all individual reactions of the individual proteins and checking whether the conditions that allow a certain reaction to take place are fulfilled. A binding reaction, e.g., may only take place if the binding site is empty, and an electron transfer can only occur when the donor is reduced and the acceptor oxidized. If the conditions are fulfilled, the probability that the reaction takes place during the next time step of size Δbc1 complex were modeled as two independent elementary reactions each with their own conditions and their own rate constants. For the regimes considered here thermodynamically driven reverse reactions can be neglected.In our approach none of the microscopic reactions was implemented as reversible. Explicitly reversible processes such as the association/dissociation of a metabolite or the back-and-forth hinge motion of the Rieske domain of the cytochrome c2 and quinone, two sites for electrons residing on the special pair and the Qb quinone, respectively, and one “counter” for the number of electron-proton pairs on the Qb. With zero, one, or two electron proton pairs on the Qb and the other four sites either empty or occupied, a single RC alone can already be in 48 different states. In a traditional state-based model all these 48 states have to be included, in principle, and updated simultaneously for a complete description of the state vector of a single RC. With more protein types and more complex models, the number of possible states will grow exponentially. For traditional approaches, rule based setup tools have been developed to help with this tedious and error-prone setup of, for example, receptor phosphorylation in signaling cascades As a consequence of our implementation from individual microscopic reactions we do not have to deal with the combinatorial explosion of the number of states which appears in traditional rate-equation approaches from treating all possible combinations of charge and oxidation states of the multiple sites in the proteins. A simple protein unit such as the reaction center can be modeled with two external binding sites for cytochrome t = 1 microsecond on a current desktop computer. This is fast enough to perform even an extensive multi-parameter optimization on a small cluster of workstations. The largest optimizations performed in this study took about ten days on 40 CPUs. This, however, is still much less time than what it took to compile the data, set up the simulation model, or to analyze the results.Even though our approach is not optimized for numerical efficiency, the fast dynamics of a complete vesicle during a flash experiment can be simulated close to realtime at a time resolution of Δbc1 complex were lumped together into two reactions c1 domain, the current version now explicitly models the different pathways of the two electrons away from the QH2 docked into the Qo site, see c1 domain and the hinge motion of the Rieske iron-sulphur domain (FeS) between the b and the c1 domains (positions FeSb and FeSc1 in bL and bH on its way from Qo (indicated by the binding site bsQo in i (bsQi). Also, the proton release into the vesicle is now coupled to the gating motion of the FeS domain (reaction D2) bH and bL, which are the major charge transfer steps against the transmembrane potential . This set of reactions reproduces the Q-cycle proposed by Crofts In our first model, most of the internal reactions of the cytochrome Further details and the implementation of the other proteins, i.e., of the LHC, the RC, and the ATPase are given in chem to a proton concentration (or pH) difference across the membrane 0 was derived from a thermodynamic analysis at a planar wall separating two infinitely large half spaces chem = 200 mV. To make as few assumptions as possible about the driving force of a proton gradient in this small vesicle, we treated ΔΦ0 as a freely adjustable conversion factor that relates the concentration of free protons inside the vesicle to a contribution ΔΦchem to ΔΦ. ΔΦ0 was then tuned in the optimization such that the kinetic response of ΔΦ matched the experiments as well as possible.The second modification of our earlier model averaged fractional proton numbers in the vesicle, eliminating the unrealistic discretization steps in the pH difference due to the very small number of free protons. To account for such titratable groups inside the vesicle, a special pseudo-protein was included in the simulations which has Np proton binding sites with a given pK value. Each of these Np proton binding sites is protonated with a probabilityNp = 80 for our model chromatophore. Assuming, for example, a pK = 4.5, which is close to the pK = 3.8 value of aspartic and glutamic acid residues which are often found on solvent exposed protein surfaces, 76 protons are then required to generate ΔΦchem = 200 mV. Only four of these 76 protons will be free in solution whereas the other 72 are bound to the titratable groups. In the optimization (see below), pK was varied to check for its effect. Tests showed that for Np>80 the dynamic behavior of the vesicle did not change anymore. Also, as will be shown below, for increasing Np the importance of ΔΦchem actually decreases relative to the electric contribution ΔΦcap which accounts for the charges of the protons themselves.In real biological vesicles, however, the lipids of the membrane and the solvent exposed protein surfaces contain a large number of titratable groups. These allow for chem this contribution was calculated from the total number of charges Nq and a freely adjustable conversion factor ΔU0 for the transmembrane potential created by a single charge.U0 describes the (yet to be determined) effective inverse capacitance of the complete vesicle without any assumptions about its form or dielectric properties. For comparison, Feniouk et al. used the specific capacitance per area of 1 µF cm−2 of a planar membrane to estimate the electrical capacity of a typical vesicle as 5×10−2 fF U0 = 3.2 mV/e and the 76 H+ estimated above for ΔΦchem = 200 mV lead to an additional ΔΦcap≈240 mV.This second contribution to ΔΦ arises not only from the charges of the protons inside the vesicle but also from those bound to the titratable residues and from the charges displaced perpendicular to the membrane in the proteins. Similar to ΔΦcap and ΔΦchem was reported by Klamt et al., who found that two thirds of the protonmotive force are due to the charges and one third stems from the proton density gradient cap is much more important for the chromatophore kinetics than ΔΦchem (see below). Here we did not attempt to fix the conversion factors ΔΦ0 and ΔU0 from a physical model but tried to find values for them that allowed reproducing the time dependent experiments in the best way. Only then can we interpret the obtained results in thermodynamic terms.Based on observed relaxation times, Feniouk et al. concluded that the chemical proton buffering capacitance is about 3.5 times larger, i.e., less important for ΔΦ than the electric contribution. A slightly smaller ratio between ΔΦchem increases logarithmically with the number of free protons, ΔΦcap grows linearly with the total number of charges. Consequently, for a few protons in the vesicle, ΔΦchem is the main contribution, while for larger numbers of protons or charges displaced in the activated proteins, ΔΦcap dominates and the accurate description of ΔΦchem becomes less important. For the bacterial ATPase, it does not matter whether the proton driving force is generated via ΔΦcap or via ΔΦchemcap matters. With the numbers estimated above, we can consequently expect that for metabolically relevant values of the total transmembrane potential ΔΦ = ΔΦchem+ΔΦcap≈200 mV, where the ATPase already runs at full speed, the dynamic response of the photosynthetic chain is mainly determined by ΔΦcap.Due to their different physical origins, the two terms contributing to ΔΦ scale differently with an increasing number of protons in the vesicle. While the chemical contribution ΔΦThe optimization of the rate constants was started from a set of rates compiled from experimentally determined reaction rates and estimates from the steady state reconstruction Rb. sphaeroides reported by Barz et al.Rb. sphaeroides are not photosynthetically competent, the bacteria were grown semiaerobically and then incubated anaerobically in the dark before the actual experiments started. Some experiments were performed on dark adapted vesicles which had been extracted from cells disrupted in a French press. For all simulations, both of the whole cell and of the vesicle scenarios, we used the same standard vesicle of 45 nm outer diameter with ten dimeric core complexes composed of one LHC and two RCs, ten bc1 dimers, and one ATPase. The model vesicle also contained as carriers 200 quinones and 20 cytochrome c2 as well as 80 titratable residues. All scenarios were started from a dark adapted initial state. Consequently, the cytochrome c2 pool was initially completely reduced and only 15 of the 200 quinones were oxidized. To get reproducible statistics, the single- and multi-flash simulations were repeated 40 times for every parameter set and their outputs were averaged before running the analysis.To determine the unknown kinetic parameters, we ran molecular stochastic simulations according to seven time dependent measurements on the photosynthetic apparatus of c throughout the vesicle interior and of the quinones across the complete membrane area are much faster than the respective binding and unbinding rates We now shortly describe the selected experiments and point out how special features of the scenarios were implemented in the simulations. The letters A or B are used to denote whether the experiment was reported in A7_ΔΦ and A7_cytc: Figure 7 of c oxidation state after one single-turnover flash in dark-adapted anaerobic bacteria. The purpose of this experiment was to show that PufX is not an integral part of the electron transport chain as even without PufX cyclic electron transport can occur. Consequently, the experimental results were basically the same for wild-type and mutant strains. The experiment was performed on anaerobically incubated dark adapted cells, which we modelled by a single dark adapted vesicle. In the simulation, the cytochrome c oxidation was determined as the total number of oxidized c2 and of oxidized c1 domains of the bc1 and rescaled for comparison with the arbitrary units of the experimental data. The electric part of the transmembrane potential was measured via absorption difference spectra of the carotenoids of the LH2 cap and the cytochrome c oxidation state. In the simulation also the chemical part of ΔΦ was included. The flash was simulated by a light intensity of 1500 W/m2 for 0.1 milliseconds, which is fast and strong enough to initiate a single electron transfer from the special pair to Qb in nearly all RCs. Note that due to the stochastic nature of the simulations there is always the chance that one of the RCs does not trigger even upon an extremely strong flash. At the chosen intensity of the single flash, most RCs end up in the semiquinone state with a single electron-proton pair on the initially oxidized Qb. The total simulation time was 0.3 seconds using a timestep of 2 microseconds where the flash occurred after 30 milliseconds. The 40 independent runs per parameter set took about 50 seconds to simulate and analyze as described below.A8_ΔΦ and A9_cytc: While PufX does not affect single turnover electron transport, it is required for continuous turnover of the photosynthetic apparatus as demonstrated by the next experiment. The two quasi steady state measurements shown in figures 8 and 9 of reference cap and the cytochrome c oxidation state of intact anaerobic cells during a nine seconds long continuous illumination. Without PufX, ΔΦcap and the cytochrome c oxidation level shortly rise upon illumination and then decay again as in the single flash experiment A7, whereas in the wildtype both signals saturate after about two seconds while the light is on. Again, ΔΦcap was rescaled to the first fast step and the cytochrome c oxidation given in arbitrary units. Here, a single closed vesicle was simulated for 12 seconds using a timestep of 20 microseconds. After one second in the dark the illumination was switched on for nine seconds with an intensity of 90 W/m2. This intensity is higher than the reported value of 50 W/m2 in the experiment to ensure that the light is still saturating when the absorption cross section of the LHCs is reduced during the optimization. The five simulation runs for every parameter set took about 40 seconds including analysis.B1_Q: To show that the RC is fully functional in a PufX deficient mutant, Barz et al. probed isolated chromatophores with a series of four single-turnover flashes spaced 1 s apart b quinone bound to the RC. When loaded with a second electron-proton pair after the next flash, the quinol detaches from the RC and is replaced by another quinone. As this exchange is fast compared to the spacing between the flashes, semiquinone oscillations can be observed spectroscopically. In this experiment the bc1 complexes were inhibited by antimycin A, which blocks electron transfer from heme bH onto the quinone bound at the Qi site. Additionally, N,N,N′,N′-tetramethyl-p-phenylenediamine was added in the experiment, which efficiently reduces the RC special pair and oxidizes Qb− with a slow time constant of minutes that is negligible here. This scenario, in which only the state of the RCs was probed, was simulated with a vesicle which only contains the LHCs and the RCs. The pool of reduced cytochrome c2 was set to a constant concentration to mimick the fast re-reduction of the special pair after each of the flashes. In the experiment the chromatophore density was adapted such that each flash induced a single turnover in about 90% of the RCs. For a similar turnover probability in the simulation the flash was set to 4000 W/m2 for 0.2 ms.B6_P and B6_cytc: These two scenarios are again different measurements performed during the same experiment, where the oxidation states of both the special pair bacteriochlorophylls of the RCs (P) and of the total cytochrome c content in anaerobic dark-adapted cells were monitored during a train of 16 flashes spaced 20 ms apart  values are listed in Together with the numbers of the proteins, the geometric properties of the vesicle, and the initial conditions, i.e., the initial pool occupation numbers, the model contains 44 rate constants and parameters relevant for the kinetics. Out of these 44, we selected 25 parameters for the optimization. These includes all parameters of the LHC and the RC, respectively, 13 of the 19 parameters of the si was determined for each of the eight scenarios from the normalized squared distance between the simulation results x(ti) at equidistantly spaced time points ti and a fit function to the experimental data at the same time points f(ti):Ci were adjusted such that the scores for the best parameter sets were on the order of one for each of the experiments. However, their actual values are not important, as the same Ci were used for all parameter sets. They only affect the overall scaling of the scores without changing their relative ordering. For scoring the individual simulations, we used a fit to the experiment rather than the raw experimental data points. This was done for two reasons. First, the experimental data is noisy, i.e., each of the discrete data points has an unknown deviation from the correct value. Continuous fit functions composed from constant, linear, and exponential terms allowed us to average a smooth curve through the noisy data. Second, the data points are often measured at rather large or non-equal time intervals. With the continuous fits the simulation could be scored at the most convenient time intervals. The scoring functions for the chosen experiments are listed in supporting The quality of a given parameter set was judged from the overall agreement between the simulation results and the experiments. For this, an individual score S of a parameter set was then determined as the product of the individual scores si. Multiplying the individual scores ensures that a parameter set has to perform well in all simulations in order to achieve a good overall score in the optimization.The overall master-score P> and variance σ of a parameter P from the 1000 highest-scored parameter sets among the 32800 different parameter sets:min/Pmax.The optimal value for each individual parameter was then determined by computing the logarithmic average <bc1 dimers could suffice to supply the ATPase with enough protons to run at full speed. From wet lab experiments, however, a ratio of about one bc1 dimer per dimeric core complex was found, i.e., about ten bc1 complexes per vesicle bc1 can only use the limited amount of energy stored in the electron proton pairs on the QH2 to pump the protons against the transmembrane potential ΔΦ, its throughput decreases with increasing ΔΦ. Thus, in a steady state scenario it does not make a difference whether the model includes only three or ten or even more bc1 complexes as their turnover will be reduced once a certain transmembrane potential is reached. In a dynamic scenario, however, the response of ΔΦ to a rapid increase of the illumination will be faster in a vesicle carrying more bc1 complexes.In the previous reconstruction of a typical chromatophore vesicle bc1 dimers affects the non-equilibrium dynamic response of a chromatophore vesicle during the single flash experiment A7 introduced above cap to the transmembrane potential and the oxidation state of all cytochrome c after a short flash in anaerobic cells of Rb. sphaeroides PUFC/g mutants, which resemble the wild type  is rather irrelevant for a good agreement between the experiment and the respective simulation in this fast transient scenario. Instead of the usually used 59 mV/ΔpH, the values ranged between 1 and 20 mV/ΔpH here. With these values, ΔΦchem contributes less than 30% to the total ΔΦ. In fact, simulations that completely neglected ΔΦchem, i.e. ΔΦ::ΔΦ0 = 0, could reproduce the experimental data with comparable accuracy. As ΔΦ::ΔΦ0 increases with Nbc1 from too small towards more realistic values, one would choose Nbc1 as large as possible based on the behavior of this parameter, too.The large standard deviations for ΔΦ::ΔΦbc1::Φ0, regulates at which value of ΔΦ the pumping rate of the bc1s is reduced. Its optimal value decreased with increasing Nbc1. However, with the obtained values of bc1::Φ0≥80 mV, the throughput of the bc1s is only limited at larger ΔΦ while still allowing for a fast proton pumping at low ΔΦ. To prevent ΔΦ from rising too high during steady state situations, a smaller bc1::Φ0 should be preferable, again suggesting to use a larger Nbc1. The pK value of the titratable groups, finally, depends only weakly on Nbc1. Its trend to increase with Nbc1 can again be understood as increasing the effective buffering capacity of the vesicle by an earlier onset of proton binding.The third parameter, bc1 complexes for the typical vesicle in the further optimizations of the kinetic parameters of the photosynthetic apparatus. This value obtained from the dynamic simulations fits well with the stoichiometries of approximately one bc1 dimer per two RCs determined in wet-lab experiments Based on these results, we chose to use ten cytochrome bc1 complexes , which could be varied over more than two orders of magnitude while still reaching scores above 0.068. Consequently, the actual value of this parameter plays a minor role for a good agreement between simulation and experiment. Panel C of bc1::ktr(FeS:c = >b), a parameter of intermediate sensitivity. Here, scores above 0.068 were obtained with parameter values varying within about one order of magnitude. In each case, the peak of the distribution of the master score gives the optimal value of this parameter, and the width of the distribution encodes the importance (sensitivity) of this parameter for a correct in silico description of the experiments.To determine the optimal kinetic parameters for the individual reactions, the obtained multi-dimensional distributions of the master-scores were projected onto each of the individual parameters as shown in P> and their sensitivities Pmin/Pmax are listed in Pmin and Pmax are very close to each other and the ratio Pmin/Pmax is close to one . For the complementing observation A8_ΔΦ, the differences between simulation and experiment were more pronounced. In the simulation, ΔΦcap increased with two phases as in A7_ΔΦ and then saturated within about one second as expected. In the experiment, ΔΦ increased slower and then even began to decay again while the light was still on. Barz et al. noted that they could not explain this slow transient overshooting and they reported that ΔΦ finally reached a steady state level after some 20 seconds of continuous illumination In the quasi steady state scenarios of B6_cytc and B6_P  = 1.4 * 108 nm3 s−1 translates into a proton uptake rate of 14 s−1. Consequently, at steady state the 20 RC can reduce 125 QH2 per second which then allow the bc1 to pump 500 H+ s−1. Even under strong illumination this is only slightly more than the maximal turnover of the ATPase of 400 H+ s−1.c2 two experimental estimates of 270 s−1−1−1 indicating that the c2 are only loosely bound to the RCs. The corresponding binding rate of 9.2 * 105 nm3 s−1 ensures that even when nearly all c2 are oxidized the supply with electrons from the reduced c2 does not become rate limiting.For the unbinding of the oxidized cytochrome −1 is about twice as fast and the binding of the Q is even faster with an 80% reduced quinone pool.In the steady state reconstruction the unbinding of the QH2 from the RC was estimated to take 25 ms BC1: For the bc1 dimers one of the optimization criteria was that the steady state throughput at vanishing ΔΦ is close to the experimentally determined value of about 75 c2 reduced per second bc1 dimers could consequently pump up to 1500 protons per second into the vesicle. For the bc1, the throughput-limiting reaction was found to be the unbinding of the oxidized Q from the Qo site with a bc1::koff(Q@Qo) = 28 s−1. We also found that both the binding and the unbinding of the quinols is two to three times faster at the Qi site than at the Qo site. Additionally, most of the Q directly hop over from the Qo to the Qi site and only a few of them unbind from the bc1 back into the bulk. The same behavior was found for the QH2 generated at the Qi site indicating that the direct transfer between the Qi and the Qo sites of the dimers is important for an efficient turnover as it effectively increases the local density of substrates. When any of these rates related to quinone dynamics at the Qo and Qi sites was scanned individually, the score changed only very little or not at all with the parameter value when the other rates were close to their optimal values. This occurs because, for example, Q can arrive at the Qi site either from the Q pool in the membrane or from the nearby Qo site of the other monomer. If either of these two paths is shut down due to a too low rate constant, the other path takes over and the bc1 continues to work. In the systemic multi-parameter optimization, however, such extreme settings are less favourable because they are more sensitive to small parameter changes and therefore effectively suppressed. This is why we obtained meaningful rates even for these correlated parameters.c2 association and dissociation rates at the bc1 to the same values as at the RC, but the optimized rates are much faster. The binding of the c2 to the bc1 was about one order of magnitude faster than at the RC and the unbinding was found to be three times faster.As an initial estimate we had set the cytochrome b and the c1 domains associated with the electron transfer to the c1 heme. Here we found that the “forward” swing from the b to the c1 domain was about 30% faster than the corresponding motion back to the b domain. One can speculate that this is due to the proton release into the vesicle interior, which is gated by the Rieske domain, i.e., that when there are two protons waiting to be released they are pressing against the closed gate accelerating the opening conformational change.Our model explicitly includes the conformational change of the Rieske group between the PR: We found that some representation of protonatable residues (PR) must be present in the simulation. The optimal pK was about 5, but it could be varied between below 4 and 6.5 without any noticable differences in the time courses. Also their number was uncritical as long as it was above some 60 or 70 residues. This means that we considered on average a single protonatable group per protein. More protonatable residues would also mean that more protons can be pumped into the vesicle for the same averaged pH increase. Their charges, however, are not compensated for and the weight of ΔΦcap relative to ΔΦchem would increase further.ΔΦ: Interestingly, the optimization found an effective electric capacitance close to the expected values, whereas the chemical contribution was found to be rather unimportant for the small vesicle with its protein-filled membrane. The small spread of good ΔΦ::U0 values emphasizes the relatively high importance of the charges for a faithful representation of ΔΦ, while in our simulations the chemical contribution from the proton density difference could have even be neglected completely.0 analogous to the distributions of the master score shown in Pmin/Pmax equals 0.336 from the 1000 best parameter values. Such a broad distribution with a low Pmin/Pmax ratio denotes that this experiment is less important for the overall parameter determination. This correlates with the fact that buffering capacities do not play an important role under steady state conditions. However, a constant illumination of nine seconds length is not yet a true steady state. Indeed, the complementing measurement A9_cytc, which has the same illumination profile, shows a preference for values of ΔΦ::U0≈10 mV/e with a higher sensitivity of 0.667. In the third example, B1_Q, good scores are only obtained with ΔΦ::U0 close to its optimal value. This comes somewhat unexpected because this experiment with its wide-spaced flashes and the focus on the QB dynamics does not seem to probe the proton buffering capacity of the complete vesicle. Actually, with a value of 0.690, the sensitivity from the best 1000 parameter values is only slightly larger than for the apparently broader distribution from A9_cytc.As already mentioned above, experiments probing different setups and different kinetic regimes are sensitive to different parameters. This is illustrated in Pmin/Pmax of all relevant parameters. The resulting scores, listed in For each of the scenarios we determined an “importance score” by summing up the sensitivities c and of the special pair decay faster than in the experiments. These simulations are also sensitive to the peak value of the intensity which we could not determine from the description of the experimental setup. The best we could do was to estimate the intensity based on the verbal description as “single turnover flashes”. The same uncertainty applies to the steady state scenarios A8_ΔΦ and A9_cytc. Here the intensity was described as “saturating” which is true for any intensity above some 20 W/m2. However, the steady state is reached faster with a higher light intensity and this would have been a way to tune the results of A8_ΔΦ and A9_cytc.One of the major uncertainties when setting up the simulations was the exact time course of the illumination during the single- and multiple-flash scenarios. We used a profile where the light was switched on and off instantaneously. In reality, however, the brightness of a flash bulb increases and then decreases again continuously with possibly two different time constants. This may explain why in the simulated flash experiments  the oxidation states of the cytochrome 2 W−1 s−1Starting from the optimized parameter values of Another source of deviations is that we only simulated a single chromatophore, i.e., the parameters and simulation conditions were exactly the same for all repeated runs, whereas the experiments were performed on a collection of similar but non-identical chromatophores. To correctly account for the variations found in a real bacterium, one could run combined simulations with different chromatophore configurations, i.e., vesicles with different stoichiometries, sizes, and possibly leaks, using the same values for the individual rate constants. Along this line, one could in the future also investigate the effects of variations of the rate constants from protein to protein.Nbc1 had shown that varying the relative stoichiometries can easily change the dynamic behavior of the model. On the other hand, when the complete system is scaled such that all copy numbers are, for example, doubled, and the pools sizes are rescaled by the same factor, then the dynamic behavior remains unchanged. When only the vesicle diameter is increased then the surface area grows quadratically with the diameter and the inner volume with the third power. Consequently, the copy numbers of the transmembrane proteins and the volume of the quinone pool would increase with the same ratio, whereas the volume of the cytochrome c and proton pools grows faster. To recover the optimized binding probabilities, which are determined from the product of the association rate and the number of metabolites per pool volume, the association rates for protons and cytochrome c have to be rescaled accordingly. The rates for dissociation or internal conversion reactions remain unchanged because they do not depend on the respective pool volumes.Closely related is the issue that the rate constants were optimized for a fixed vesicle size and stoichiometry. The determination of Rb. sphaeroides on time-scales from milliseconds up to a few seconds. Processes that only occur on longer time-scales were not included in the model such as regulatory effects on gene expression due to adaption to changes in the environmental conditions as, e.g., from aerobic to anaerobic growth. Also not considered were charge or redox relaxations in response to external buffers. Such exchange reactions have to be much slower than the turnover of the photosynthetic apparatus in order not to degrade its performance by acting like a shortcut to proton or redox gradients. However, an externally set redox poise in the experiments was considered in the simulations via appropriate initial redox states of the cytochrome c2 and quinone pools. Scenario A8_ΔΦ allows to derive an estimate for the time scales of such redox or charge relaxation processes. In this experiment ΔΦcap overshoots and relaxes again within less than one minute. For comparison, the ATPase can phosphorylate about 6000 ADP molecules during this time. Unfortunately, the description of the experiment does not state whether this is a reversible or an irreversible relaxation process.Our model is also incomplete in a kinetic sense as this study is focussed on the dynamic behavior of the photosynthetic apparatus in anaerobic cells of Rhodobacter sphaeroides that the kinetic parameters for a dynamic model of a metabolic system can be reliably determined by simultaneously fitting the results from stochastic dynamics simulations against a number of time-dependent experiments. After determining an optimal set of kinetic and biophysical parameters for the chosen experiments with an evolutionary algorithm, our molecular stochastic simulation model can now reproduce the observed dynamics over a wide kinetic range from millisecond long single-flash experiments up to quasi steady state conditions. For this, we included 25 of the 44 parameters of the model in the optimization procedure. The few remaining deviations can be traced back to simplifications of the model and missing information about the experiments. With this approach we have thus successfully combined a detailed microscopic model built from molecular biological data and a systems biological parameter optimization by comparing the completely assembled system to macroscopic experimental observations. To be able to reproduce the time dependent experiments we first had to amend our model which had worked well for steady state scenarios before. The two main changes with respect to our previous model were the treatment of the transmembrane potential with the explicit contribution of the charges of the protons and the much more detailed model of the bc1 dimer which now allows to probe its internal states, too. After the systemic optimization we could then interpret the obtained optimal parameter values in the molecular realm. Thus, the two approaches were connected in both directions: molecular data lead to a systemic description and the behavior of the complete system identified important components of the molecular description.We have shown for the well understood model system of the photosynthetic apparatus of the purple bacterium a priori whether a parameter determination would actually succeed in simultaneously narrowing down such a large number of parameters so that a reasonable agreement between the experiments and the simulation could be achieved. In fact, the evolutionary optimization found reasonable parameter sets already within the first ten iterations. During the next 30 iterations, the quality of these parameter sets was further increased. The evolutionary optimization not only gives the best parameter values but also, via the spread of the “good” solutions, allows to judge the importance of each parameter and of each experimental scenario used for the comparison. In the largest optimization run 54400 parameter sets were tested, optimizing 25 parameters simultaneously. The same number of data points on a 25 dimensional grid would mean less than two grid points per dimension. Consequently, a systematic scan for appropriate parameter values over intervals spanning several orders of magnitude each would have been absolutely impossible. On the other hand, the convergence of the scores when optimizing 25 parameters was not much slower than the convergence of the next smaller runs with 15 and 12 variable parameters, respectively. This indicates that optimization runs appear feasible using this approach where up to 25 parameters are optimized simultaneously. We suggest that the underlying fitness landscape has a funnel shape such as for protein folding which may explain why the stochastic dynamics of the evolutionary optimization successfully recovered the “native state” in such a high-dimensional space. The most efficient strategy for systems where less prior knowledge is available than for the very well-studied bacterial photosynthesis is therefore to first search within really wide parameter ranges and then iteratively confine the search space to those parameter regions where good scores were obtained. An evolutionary optimization algorithm quickly finds “good” solutions but the convergence is relatively slow. Therefore, this iterative refinement of the parameter ranges is more efficient than trying to obtain converged results from a single optimization step.Remarkably, only about one third of the stoichiometries, rate constants, and parameters of the model, which are all related to the kinetic behavior of the chromatophores, have sofar been determined experimentally. For the others only estimates were available. It was therefore not clear In total we spent more than 10 single-CPU-years of computing time on the optimizations presented here. For such a simple system, this may sound like a prohibitive amount of resources. However, with the experience from this project subsequent optimizations can be performed much more efficiently. For example, we used a very conservative short time step and relatively long initial equilibration phases. The last optimizations that we performed during this project could already be performed about three times faster than the first tests. Also the simulation code can be further optimized or parallelized to make efficient use of GPUs. Consequently, at least ten times larger systems can be optimized already now and even larger ones in the near future. However, for these larger and more interesting systems the bottleneck will not be the computational resources but the amount of available experimental data which is required to parametrize the many kinetic constants. Therefore it might actually be a more promising approach to parametrize the enzymes individually before they are plugged together. Such optimizations can already now be performed overnight on an average desktop computer.N/5 independent kinetic experiments to be able to narrow the room of solutions for a cellular system with N reactions to a single basin of attraction. More work on other systems is certainly required to substantiate this claim.The good agreement is remarkable also when considering that the experiments used as reference were not designed for a quantitative analysis of the photosynthetic apparatus but to elucidate the role of a specific protein, PufX. They were therefore not all performed under the very same external conditions. Also, some basic details such as the light intensities inside the samples or the actual intensity profiles of the flashes, which are crucial for the simulations, were not reported because they were not relevant for the questions asked. Based on the current results we expect that an even more reliable parameter determination and a more stringent verification of the simulation model could be performed with a set of experiments specifically designed for reproducible, quantitative measurements. Some of these experiments would be routine tests like applying the same dynamic illumination profiles under varying external redox or pH conditions. For the optimization presented here, only four different observables were measured, but over a wide range of timescales. This then allowed to reliably determine even parameters that are some reactions steps away from the measured quantities. It is therefore crucial that the set of experiments covers a wide range of the kinetic regime accessible to the system under consideration. Based on the experiences of this study, we suggest that one needs about During the optimization we did not find any unexpected parameter values, which would have indicated that our model was partly incorrect or not complete enough. For the wellknown bacterial photosynthesis we could set up the proteins from agreed upon biochemical models, but for less thoroughly studied systems the implementation might actually require the decision which of the available models to choose. Then, such a systemic parameterization can be also used to test which of the presented models behave correctly and potentially even to figure out why a certain model fails.bc1 complex and the quinones. Most importantly, we obtained values for the initially missing two thirds of the kinetic parameters of the model. For the bc1 complex we found that the quinones and quinols are exchanged preferentially between the two halves of the dimer in a local micro-environment with more favorable effective quinone oxidation states than in the bulk. However, the model of the bacterial photosynthetic apparatus as presented here is by no means complete. The design citeria for the individual proteins was to be able to reproduce the selected experiments which focus on the enzymatic functions of the proteins. Explicitly excluded are therefore all regulatory processes, while some of the very fast exciton and electron kintics inside the proteins could be modeled sufficiently well with effective reactions. When new experiments are added to the current set it is straightforward to update the current protein “templates” to the then required level of detail and thus to test these improved models for different observables and on a wider range of timescales.The obtained fully parametrized model gave important biological insights about the complete electron transport chain, especially about the central cytochrome Rb. sphaeroides to show that molecular data can be compiled with our pools-and-proteins approach to obtain consistent systemic answers. In this case of a well understood biological system wrong or inappropriate results can be identified a posteriori. However, the success indicates that the methodology can also be applied to other more complex or less well understood cellular subsystems. The next logical step at a higher level of complexity would be the bioenergetic processes of an entire bacterium or mitochondrium. Also signalling processes are good candidates for such a bottom-up molecular stochastic description together with the systemic evolutionary parametrization. In the case of the respiratory system, some of the proteins, namely the cytochrome bc1 complex and the ATPase, can be taken unaltered from the current model of photosynthesis together with their optimized parameters. Modeling and parametrization of other biological systems can thus be started from the already known proteins. By this, a library of protein models can be built up, with which finally the complete metabolism of an entire cell could be simulated. The resulting models at the molecular scale can also be used to test or verify under which conditions simplifications may be applied to reduce the complexity of larger models without changing their dynamic response. Correspondingly, we expect exciting progress in this area in the near future.Here we used the well established photosynthetic apparatus of http://service.bioinformatik.uni-saarland.de/vesimulus free of charge for academic use.The molecular stochastic simulation framework with all currently implemented proteins can be obtained from the authors at N parameter sets. For this, the parameter values were distributed equally on a logarithmic scale within a priori chosen boundaries. From this initial generation of parameter sets, a set of input files for the simulation engine was generated from the template input files. Each of the templates describes one simulation setup corresponding to one of the experiments. Then, for each individual parameter set the set of stochastic simulations was run and the master score was determined as explained in the N/4 parameter sets of the current generation with the best scores were retained unchanged  copies of the best individuals by randomly changing each of the parameters considered in the optimization within a range of ±35%. The next N/4 new parameter sets were “cross-bred” by randomly selecting two sets from the best and taking the arithmetic average of their parameters (operation “mix”). The remaining N/4 slots were filled up with randomly initialized parameter sets. Again, a set of input files for the simulations was generated from the templates, and the parameter sets were scored by running and analyzing the respective simulations. This mutation-scoring-selection process was iterated until the scores converged. Limits for the parameter values were imposed only for the random initialization, but not for the subsequent mutations where they were allowed to take arbitrary values. However, in the optimizations performed here, the optimal values for the parameters were always found within the a priori estimated intervals.The automated parameter optimization was performed with an evolutionary algorithm as introduced by Rechenberg global from any of these already considered parameter sets. This distance δ was calculated as a normalized Euclidian distance between the vectors of parameter values p1 and p2:global of any of the previously scored parameter sets, it was discarded and replaced by a new randomly chosen parameter set—which was again tested. This approach avoided rerunning the relatively expensive simulations for a region of the parameter space that had already been examined previously. For the mutations, no such distance criterion was applied to allow for arbitrarily small improvements.To increase the convergence of the evolutionary optimization and to prevent that a complete generation gets stuck in a single local optimum, we extended the standard algorithm by two distance constraints between the parameter sets. For the first global constraint, all parameter sets that had been considered so far, including the ones in the latest generation, were saved in a global history. Each newly generated parameter set, be it from the averaging or from the random initialization, then had to differ by a predefined distance δN/4 from a generation. Starting from the parameter set with the best score, the next best set was only considered when its parameter vector was further away than δlocal from any of the already selected sets. If not, it was skipped and the next best parameter set was tested until enough good parameter sets were collected. With this criterion we could avoid that all parameter sets chosen for the next iteration were located in the same local optimum. Obviously, δlocal should be chosen at least as large as δglobal. The benefits of using these two criteria were confirmed by testing them with specially crafted test cases which contained multiple maxima of different widths and heights. Currently, the minimal distances between two parameter sets are independent of the actual parameter values or the scores achieved in a certain region of the parameter space. In a more sophisticated implementation, they could be related to the scores of the parameter sets and increase the minimal distance for low scores, while allowing for smaller distances where the scores are good. Thus, the regions of parameter space where the experiments cannot be reproduced at all would be sampled more sparsely than the interesting regions.A second local distance constraint was applied when selecting the best Because all parameter sets of a given generation are scored independently, the parameter optimization with an evolutionary algorithm can be conveniently parallelized by running the simulations for different parameter sets on different nodes of a compute cluster.c2 from the RC experimental values were available around which the interval could be centered. Often, a lower limit for a parameter could be estimated from the steady state throughput of the respective protein. Alternatively, we ran tests where a single parameter was varied manually to get an estimate of the required range. With this first set of intervals the two main parameter optimization runs were performed. These ranges are listed in For the random initialization of new parameter sets we had to specify a range for each parameter to be optimized. These intervals should be wide enough so that the optimal parameter value is inside the interval. On the other hand, the convergence of the optimization is faster when a smaller interval is used. For the well known bacterial photosynthesis we estimated the search ranges from various sources of information. For some of the parameters like the dissociation constant of the To validate that the chosen intervals, which for some parameters spanned only one order of magnitude, were indeed wide enough, we also ran optimizations with very wide intervals for the initial values which spanned four to six orders of magnitude see . With thFigure S1bc1 Complexes: Evolution of the master score.Determining the Number of (0.07 MB PDF)Click here for additional data file.Figure S2bc1 Complexes: Average Master Score vs. the number of bc1 complexes Nbc1.Determining the Number of (0.09 MB PDF)Click here for additional data file.Figure S3Evolution of the Scores for Different Subgroups.(0.10 MB PDF)Click here for additional data file.Figure S4Scores from Some Experiments During the Scan of One Parameter.(0.09 MB PDF)Click here for additional data file.Figure S5Comparison of a Score Distribution Obtained from a Main Parameterization Run and from a Wide Range Verification Run.(0.98 MB PDF)Click here for additional data file.Table S1Model Parameters Not Included in the Optimization Process.(0.08 MB PDF)Click here for additional data file.Table S2Sensitivities of the Experimental Scenarios.(0.08 MB PDF)Click here for additional data file.Table S3Search Ranges for the Parameter Optimization.(0.08 MB PDF)Click here for additional data file.Text S1Fit Functions to the Experimental Time Traces.(0.24 MB PDF)Click here for additional data file.