Monte Carlo analysis of an ODE Model of the Sea Urchin Endomesoderm Network
- Clemens Kühn^{1, 2},
- Christoph Wierling^{1},
- Alexander Kühn^{1},
- Edda Klipp^{2},
- Georgia Panopoulou^{1},
- Hans Lehrach^{1} and
- Albert J Poustka^{1}Email author
https://doi.org/10.1186/1752-0509-3-83
© Kühn et al; licensee BioMed Central Ltd. 2009
Received: 19 May 2009
Accepted: 23 August 2009
Published: 23 August 2009
Abstract
Background
Gene Regulatory Networks (GRNs) control the differentiation, specification and function of cells at the genomic level. The levels of interactions within large GRNs are of enormous depth and complexity. Details about many GRNs are emerging, but in most cases it is unknown to what extent they control a given process, i.e. the grade of completeness is uncertain. This uncertainty stems from limited experimental data, which is the main bottleneck for creating detailed dynamical models of cellular processes. Parameter estimation for each node is often infeasible for very large GRNs. We propose a method, based on random parameter estimations through Monte-Carlo simulations to measure completeness grades of GRNs.
Results
We developed a heuristic to assess the completeness of large GRNs, using ODE simulations under different conditions and randomly sampled parameter sets to detect parameter-invariant effects of perturbations. To test this heuristic, we constructed the first ODE model of the whole sea urchin endomesoderm GRN, one of the best studied large GRNs. We find that nearly 48% of the parameter-invariant effects correspond with experimental data, which is 65% of the expected optimal agreement obtained from a submodel for which kinetic parameters were estimated and used for simulations. Randomized versions of the model reproduce only 23.5% of the experimental data.
Conclusion
The method described in this paper enables an evaluation of network topologies of GRNs without requiring any parameter values. The benefit of this method is exemplified in the first mathematical analysis of the complete Endomesoderm Network Model. The predictions we provide deliver candidate nodes in the network that are likely to be erroneous or miss unknown connections, which may need additional experiments to improve the network topology. This mathematical model can serve as a scaffold for detailed and more realistic models. We propose that our method can be used to assess a completeness grade of any GRN. This could be especially useful for GRNs involved in human diseases, where often the amount of connectivity is unknown and/or many genes/interactions are missing.
Keywords
Background
Today, experimental research has uncovered a great amount of regulatory interactions between different transcription factors (TFs). These interactions can be summarized in Gene Regulatory Networks (GRNs) that control the differentiation, specification and function of cells at the genomic level. The levels of interactions within large GRNs are of enormous depth and complexity. Details about many GRNs are emerging, but in most cases it is unknown to what extent they properly describe a given process, i.e. the grade of completeness is uncertain. This uncertainty stems from limited experimental data, which is the main bottleneck for creating detailed dynamical models of cellular processes. Parameter estimation for each node is often infeasible for very large GRNs. These GRNs are static representations of the interactions and can provide scaffolds for fine grained low-level models [1]. A mathematical low-level model allows for a detailed quantitative analysis of the system and has predictive power. Construction of detailed quantitative models of large GRNs is often infeasible because the underlying data is too sparse to parameterize the model. Analysis of model properties on static network graphs, on the other hand, provides only limited insights.
To circumvent these shortcomings, we propose to construct a scaffold model of ordinary differential equations (ODEs). Analysis of some key properties of this model is feasible without knowledge of the kinetic parameters. The detected properties are compared to experimental data for validation. Parameterization of this model can be achieved by iteratively improving parts of the model once sufficient data become available.
The structure of the network is constructed from a multitude of perturbation experiments, including injection of morpholino-substituted antisense oligonucleotides that effectively knock down mRNA translation (KD) of a specific gene [5], injection of engrailed repressor domain fusions to turn activators into repressors, or mRNA over expression (MOE). The network as described in [4] is an interpretation of a vast amount of experimental data. As pointed out by Smith [6], perturbation experiments may lead to ambiguous results, which need to be clarified in subsequent experiments. Although the experimental results currently available are solid and only error prone due to incompleteness, we believe that a computational analysis of the Endomesoderm Network could greatly substantiate its interpretations. Additionally, the analysis of a mathematical model allows for the detection of possible errors and identification of ambiguities in a more comprehensive way.
The detailed results of our analysis show which parts of the model reproduce the experimental data exceptionally well or exceptionally poor. Additionally, our analysis shows a large number of effects not yet tested experimentally. These predictions can be used for experimental validation.
Our work provides a scaffold for quantitative kinetic modeling of the endomesoderm network, publicly available in SBML format in the BioModels database [16]. Refinement of this scaffold model is straightforward: Well studied parts of the model (e.g. the PMC region) can be refined in isolation and then reintegrated upon the scaffold. Such a stepwise improvement of the model is, in our opinion, less error prone than setting up a comprehensive detailed model from scratch. The iterative nature of the approach also provides a simple way to organize concerted efforts and directly shows which parts of the endomesoderm GRN require most attention in future experimental research.
Results and discussion
Heuristic Inference of Topological Perturbation Effects of ODE Models
In the following, we describe the method we use to detect topological perturbation effects from ODE models of GRNs and give a sample application that indicates the reliability of this method. The heuristic we propose here allows for an assessment of the correctness of a GRN topology given the underlying data without the need to estimate parameter values. It is thus applicable to very large GRNs. A diagram of the method is given in Figure 2.
If realistic kinetic parameters for a given GRN model are known, the quantitative effect of the knockdown (KD) or mRNA overexpression (MOE) of one or more of its components can be simulated. The qualitative effect of KD or MOE conditions that are invariant to the choice of parameters can be computed for large networks using our method. Since these effects are inherent in the topology of the GRN, we call them topological perturbation effects (TPE).
Extraction of TPEs from small GRNs can be straightforward and obvious, but for large GRNs, manual detection of all topological features becomes infeasible and error-prone. To detect TPEs of a given network topology, logical models can be applied [17]. The advantage of our continuous approach is that the model can be improved stepwise by restricting the distribution from which the parameters are drawn or fixing certain known parameters, eventually yielding a comprehensible quantitative model.
Given the topology of a GRN that has been constructed in some way from experimental data (step 1 in Figure 2), we construct a corresponding ODE model M_{0} (step 2 in Figure 2). For each experimental perturbation Pert_{ i }, a model variant M_{ i }is created and all model variants are simulated using different randomly sampled parameter sets P_{1..K}, yielding simulation results (step 3 in Figure 2). Differences in simulation results for a pair that are consistent under most parameter sets are detected as TPEs (step 4 in Figure 2). The effects detected with this Monte Carlo approach are then compared to experimental data to compute the agreement between experimental data and constructed model (step 5 in Figure 2).
TPEs can be detected for a wide range of perturbations to the model. In contrast to experimental cis-regulatory analysis, this computational approach can identify the effects of all possible perturbations on all genes in a proposed GRN simultaneously. This predictive power of the analysis can be used to test the proposed GRN by targeted experimental research.
Inference of the Reliability of the Proposed Heuristic
The chosen subnetwork consists of interactions between Blimp1, Bra, Brn, Eve, FoxA, GataE, Hox, Otx, Pmar1 and Wnt8. Parameters have been estimated for this model [18]. We denote the in-silico perturbation data (KM^{ est }) and the computed TPEs (KM^{ mc }).
Comparison of the in-silico data KM^{ est }, which contains TPEs and parameter-specific effects, and TPEs KM^{ mc }show the following results: 73.8% of the perturbation effects were equally detected in both settings. False negatives (effects detected using KM^{ mc }and not detected using KM^{ est }) constitute 8.1%, false positives constitute 15.7% and 2.1% of the effects are detected with opposite signs in the two settings. This indicates that our heuristic approach to identify TPEs yields satisfying qualitative results at least for small networks. If the wiring of the model is equal to the real topology, about 74% of TPEs detected with our Monte Carlo approach should match experimental perturbation data.
Assumptions Underlying the ODE Model of the Endomesoderm GRN
The method explained above is most useful when dealing with large models associated with sparse experimental data. Parameter estimation is infeasible here, but our method allows a preliminary assessment of the main features and comparison to experimental data. The endomesoderm network is a large network with sparse associated data. In order to set up the actual ODE model, we make the following assumptions concerning the relevant biological mechanisms during development and their mathematical representation. The model is concerned with the endoderm, mesoderm and PMC territories of the early sea urchin embryo. For detailed information on sea urchin development, please refer to [2] and references therein.
Although each embryonic territory differs concerning the expression of genes, abundance of transcription factors (TF) and signaling molecules, we assume that each cell contains the same genetic information, i.e. that no histone modification occurs in the early stages of development. We assume that each territory consists of a homogeneous number of cells, i.e. that cells in the same territory contain the same combination of TFs and express the same genes. If gene expression is regulated in part by extracellular gradients, these regulating gradients can be assumed to differ throughout individual territories and thus the regulated genes are also likely to be expressed differently in different parts of the same territory. But unless we simulate a whole population of cells (see [19–21] for approaches to modeling populations of cells), an ODE model is only capable of assessing averages of this population.
In our model, we will not try to reproduce expression time courses or spatial expression patterns. The objective is to detect the topological features arising from the Endomesoderm GRN and to compare them with experimental perturbation data. The assumptions stated above enable a straightforward modeling of the three territories referenced in the Endomesoderm Network: each territory (endoderm, mesoderm and PMC) is modeled as one cell, each containing the same regulatory interactions.
Important signaling events in development are controlled by intercellular signaling, either direct or by extracellular gradients. The effects of intercellular signaling are necessary to produce differential expression, one of the key aspects of development. For simplicity, we reduce the dynamic interaction between the different cells of the embryo to a static input depending on assigned cell type and simulation time. Making this simplification, we lose possible topological effects that manifest only in the interaction between cells. On the other hand, we avoid possible errors as well as excessive computation. Realistic modeling of extracellular gradients would require spatial modeling of a growing embryo. This would require simulation of a growing population of cells each containing an ODE model. The population of cells would need to contain territories not included in the Endomesoderm Network. It would further require realistic numbers of cells in each territory and subpopulation of a territory producing a specific extracellular signal and additional assumptions concerning diffusion rates.
An ODE model constructed according to the assumptions made above can produce differential expression in different territories depending on the static inputs. It can simulate the effect of perturbations within a territory. It cannot simulate the effects of perturbations that propagate through intercellular signaling unless the static inputs are modified to dynamic interactions between the different territories.
Construction of the ODE Model of the Endomesoderm Network
The assumptions made above are used to construct an ODE model from the topology of the Endomesoderm Network (Step2 in Figure 2). Since the detailed cis-regulatory interactions, and thus transcription kinetics, are not available for all genes in the network, we chose to construct a provisional scaffold model including all genes, but using simplified transcription kinetics. From this provisional model, we hope to eventually set up a realistic model, refining individual transcription kinetics by hand. For most genes in the network, the regulatory inputs and their nature are known. But the detailed cis-regulatory wiring, which would be necessary to set up realistic transcription kinetics, is only known for a very limited number of genes, for example Endo16 [22]. Therefore, we chose the following simplified approach.
The model contains three different embryonic territories, endoderm, mesoderm and primary mesenchyme. The three different territories are identical copies of the transformed Boolean model. External inputs to the different territories are modeled as Hill-functions modulated by events that are territory-specific. Initial conditions were set according to the Biotapestry view of the Endomesoderm Network and are also tissue specific. The model contains 54 genes and 140 variable species, 278 reactions and 287 parameters for each cell type (See Additional File 1 for the rate laws used in the model.).
Monte Carlo simulation of knockdown and overexpression experiments using randomly sampled parameter sets
We chose a multiplicative decrease for knockdown experiments because in the analogous experiments, the transcribed mRNA is blocked from translation similarly in case of high and low expression rates.
Overexpression in experiments was brought about by injection of synthetic mRNA, so the resulting increase in expression is independent of the endogenous expression. For this reason we chose an additive increase for overexpression.
Comparison of Experimental and Simulated Perturbation Effects
The qualitative comparison of perturbations either detected experimentally or in simulation using different sampled parameter sets (step 5 in Figure 2) show that the model does not reproduce all experimentally detected effects. The quantitative experimental data were converted to qualitative results using sensible thresholds in order to compare them to the qualitative simulation results (step 6 in Figure 2; see section Methods).
The effects of KD (and overexpression perturbations) in the simulation results are recorded for each of the three territories (endoderm, mesoderm and PMC). As the experimental data only show effects on the whole embryo and not only on a specific territory, we consider as a match the accordance between experimental data and simulated effects in at least one of the territories. This seems reasonable because we omitted the dynamics of intercellular signaling, which might lead to propagation of a perturbation from one territory to another and several genes are only expressed in one territory, both in model and experimental data. Using this scheme to compare simulation and experimental data, we find that 48% of the KD experimental data (43% including MOE) are qualitatively reproduced in our model. We note that the few MOE experiments available compare significantly inferior (only 28% matches with experimental data) to KD experiments (with the exception of Pmar1 MOEs), reflecting the unphysiological conditions that are created by injection of mRNA in the embryos, where often amounts are injected that can be several to hundreds of orders of magnitudes higher than the endogenous amounts produced.
Summarized comparison of TPEs and experimental data
Perturbation | A | B |
---|---|---|
Hnf6_KD | 9/14 | 0.6429 |
Otx_KD | 1/2 | 0.5000 |
FoxA_KD | 9/16 | 0.5625 |
TBr_KD | 4/7 | 0.5714 |
Alx1_KD | 11/17 | 0.6471 |
Gcm_KD | 7/11 | 0.6364 |
Hox_KD | 6/13 | 0.4615 |
Bra_KD | 8/12 | 0.6667 |
Dri_KD | 4/16 | 0.2500 |
SoxB1_KD | 2/5 | 0.4000 |
Ets1_KD | 2/18 | 0.1111 |
Blimp1_KD | 3/9 | 0.3333 |
Eve_KD | 3/8 | 0.3750 |
GataE_KD | 19/28 | 0.6786 |
Snail_KD | 2/2 | 1.0000 |
FoxB_KD | 0/2 | 0.0000 |
GataC_KD | 0/2 | 0.0000 |
Pmar1_KD | 0/2 | 0.0000 |
Brn_KD | 0/2 | 0.0000 |
Pmar1_MOE | 11/28 | 0.3929 |
Alx1_MOE | 5/15 | 0.3333 |
SoxB1_MOE | 7/39 | 0.1795 |
total | 113/268 | 0.4216 |
total KD | 90/186 | 0.4839 |
total MOE | 23/82 | 0.2805 |
The comparison also indicates genes that frequently react to simulated perturbations as described in experimental data (Pks, Nrl, FvMo, Alx1 and Bra) and genes for which simulation and experimental data rarely match (Sm50, Sm27 and Ficolin). This behavior might be caused by the large number of upstream interactions varying in strength with the different parameter sets. Other genes, which are important regulatory genes, like FoxB, FoxA and Eve also frequently fail to reproduce the experimental data. These genes have important regulatory roles. Therefore a major conclusion of this analysis is that the wiring of these genes deserves refinement or that the correct regulation here crucially depends on the chosen kinetic parameters.
A comparison of the agreement of simulation results with experimental data for each of the different territories with experimental data reveals that the PMC territory has the highest accordance to experimental data (endoderm: 22.3%, mesoderm: 25%, PMC: 36.5%). This high accordance is most likely due to the extent to which the regulatory interactions in each territory have been investigated. Indeed, the PMC GRN is the best studied part of the network and a recent publication by Oliveri et al. describes a complete PMC GRN [26], which we are currently evaluating on its own.
We need to note that the method presented here is a heuristic approach. Although our results indicate that the Endomesoderm Network topology is not sufficient to reproduce all experimental data, we cannot exclude the possibility that there might exist a choice of kinetic formulas and corresponding parameter values that enables the given topology to perfectly reproduce the experimental data.
In summary, these results indicate the need for a detailed analysis of the regulatory interactions involving Ets1 and SoxB1, while the regulation of other genes should be investigated in more detail as well (FoxB, FoxA and Eve). Overall, the model reproduces 48% of the experimental data (excluding MOE).
Predictions for further validation of network topology
perturbation | predicted effect on |
---|---|
Ets1-KD | Delta, TBr |
FoxA-KD | Dpt, FvMo, SuTx |
Eve-KD | FvMo, SuTx, Apobec, Brn, Not, Nrl, OrCT, Pks |
Pmar1-KD | Ets1 |
Comparison with Updated Network Topology
As the endomesoderm GRN is work in progress, new data and interpretation thereof is constantly added. An updated version of the endomesoderm GRN can be accessed at [4]. The topology underlying our investigation has been published on the web in December 2007. We compared the wiring of the genes that we described as deserving refinement (see above) with their wiring in the actual topology at [4].
We could not find changes in the wiring of Ets1, SoxB1 and FoxB. The wiring of the downstream genes noted above as reproducing experimental data poorly (Sm50, Sm27 and Ficolin) were not changed as well. We did, however find changes in the wiring of FoxA (regulatory input by Hox instead of Tgif) and Eve (regulatory input of Blimp1 removed). These and other changes in the topology will affect TPEs and their consistence with experimental data. But not all of our concerns have been addressed, like Ets1 and SoxB1, for example.
Certainly, the updated wiring needs to be addressed in future models that build on the scaffold proposed here. It is, however, out of the scope of this analysis to keep the model up to date with the newest topology at [4]. Our aim is to propose a provisional model that can serve as a scaffold for refining and updating the network and provide a benchmark for judging the topology.
Comparison with Topologically Randomized Networks
In order to evaluate and rank the agreement between simulation and experimental data, we constructed randomized networks from the Endomesoderm topology and computed their accordance with experimental data. The agreement between the randomized networks tested here and the experimental data is no higher than 23.5%, roughly half as good as between the correct ODE model and experimental data (48%). Two random networks with comparable features were constructed by randomly swapping edges in the original ODE model as described in section Methods and [27] (step 'R' in Figure 2). The randomized networks were subjected to the same method as the original network: conversion to ODE model, simulation with random parameter sets, detection of TPEs, comparison with experimental data. For the original model, using only as little as 50 parameter sets for simulation resulted in a similar agreement with experimental data as when using 800 parameter sets (data not shown). Hence, 100 parameter sets were used for the randomized models to decrease computation time. The randomized models also contained three identical submodels, which only differ in their temporal inputs. The number of edges switched in the randomization was set to 50 times the number of genes in the network, resulting in about 3, 000 exchanges of edges. The randomized models analyzed here were able to reproduce only 20.15% and 23.5% of the experimental data. We also investigated an additional randomized model in which we switched 30, 000 instead of 3, 000 edges and found similar results (data not shown).
Although this agreement with experimental data is in the range of the endoderm part of the original model, no randomized model exceeded this accordance significantly, as does the PMC part of the complete model (see Figure 9). Also, a combination of any three of the randomized networks did not yield an overall accordance with experimental data greater than 25%. We therefore assume that the computed agreement with experimental data of the randomized networks is dependent only on the general topological features of the model (like size and connectivity), the experimental data and the temporal inputs. This indicates that the agreement with experimental data between a model of comparable general features as the one described here with the specified temporal inputs and the experimental data used here can be expected to be less than 25% by chance alone. The accordance with experimental data expected by chance is thus about half the accordance observed using the original topology, indicating significantly improved validity of the Endomesoderm Network compared to random networks.
Applicability of the Method
The method applied here is generally applicable to any GRN. It is applicable without any prior knowledge of parameters, although a reasonable sampling distribution should be specified.
Using only very limited information, our method is able to extract topological features of a GRN, which can be compared to experimental data. Predictions made by the model can be checked experimentally to refine both ODE model and GRN topology.
In contrast to other, i.e. discrete modeling frameworks, the resulting model can be used for detailed revision and refinement. The refined model can be subjected to the same method for evaluation again, providing a measurement for the resulting increase or decrease in accordance with experimental data. Furthermore, once a satisfying accordance is reached, the constructed network can be used to iteratively estimate parameters of subnetworks. The goal of this improvement must be a fully parameterized ODE model that can be subjected to extended analysis methods. Although the method would require extensive computation when a large number of parameter sets is considered, our analysis shows that only a small fraction of the parameter space needs to be sampled to produce reliable results. We therefore propose to iteratively compute simulation runs and evaluate these until the results converge.
The method is applicable to rather large models for which parameter estimation is not feasible or to models in which different scenarios in the form of different topologies are to be evaluated.
In the case that no comparable models are available, the randomization of the model provides reasonable means of comparison. The computation of robustness to random parameter variations for all genes in the network is a by-product of the computational processes applied here (see Additional File 4). Given a considerable network size and a reasonable limit on parameter values, this robustness might help to identify further features of the network architecture, as described in [28].
Outlook
As mentioned earlier, the constructed model is by no means sufficient to quantitatively assess the temporal dynamics of the developing sea urchin embryo. In order to improve the model, intercellular effects need to be incorporated in a dynamical way, kinetics for transcriptional activation for each gene must be accurate and parameters for these kinetics need to be determined. We argue that due to the size of the network, improvements should be added in an iterative way, improving the reliability of subnetworks that can be plugged back onto the scaffold.
This can be done by choosing a subnetwork for which extensive experimental data are available (e.g. the PMC part [26]) and parameter estimation is feasible. The subnetwork can be examined in isolation and an improved version can be integrated into the scaffold model. Instead of randomly sampling all parameters, the estimated parameters values can be used and the number of sampled parameters is decreased, increasing the reliability of the analysis. Generally, transcription kinetics can be modified to express the detailed cis-regulatory logic including TF cooperativity or competitive binding. The probability distributions from which parameter values are sampled can be modified to restrict the values of certain parameters to smaller ranges if experimental data suggests this. Using these three approaches, the model can be refined and each iteration of the refinement can be evaluated by comparison with a previous version of the model.
Various approaches for modeling transcriptional activation (see [29] for an example) exist. Most of these general approaches lack details concerning possible cooperativity of TFs, for example by changing the availability of binding sites [30]. We therefore doubt that exclusive use of generalized transcription kinetics as described here and in [29]) are sufficient for a realistic, quantitative model of the Endomesoderm Network.
Once intracellular processes can be modeled satisfactory, the model has to be extended to ensembles of cellular models interacting via extracellular signaling factors. Examples for modeling ensembles of ODE models are given in [19, 20].
Conclusion
If we do understand a biological system, we can create a mathematical model that can reproduce experimental data. If the mathematical model fails to reproduce experimental data, we obviously do not understand all essential parts of the system. We have created a provisional mathematical model of the entire endomesoderm GRN, the first to our knowledge, and used our heuristic to test our understanding of this system. This test results in 48% agreement between simulation results and experimental data. For a correct model, we expect 73.8% agreement with our heurisitc (48% relative to 73.8% is 65%); randomized versions of the model reproduce only 23.5% of the experimental data. We thus conclude that the presented model is only partially correct and that some crucial interactions are not included.
To improve the model, we need to use more realistic transcription kinetics than the stereotypic ones used here. However, the underlying data is not sufficient to unambiguously assign realistic kinetics to all genes in the network. We hope that our provisional model can be used to assign improved kinetics stepwise, starting with genes for which the sufficient data is available (e.g. Endo16 [22]). In addition, it is clear that the regulatory interactions in the endomesoderm network are not complete. This is obvious for the genes that still contain inputs termed 'ubiquitous' (e.g. SoxB1 or Ets1), even in the updated endomesoderm GRN. Consequently, new experimental data is necessary to improve our understanding of gene regulation in the sea urchin embryo. For well established parts of the network (e.g. PMC), detailed data is required. For important regulatory genes like SoxB1 and Ets1, the missing regulators need to be determined. Model and heuristic proposed here can significantly improve the integration of new data by testing the resulting change in TPEs.
The method we described enables a heuristic assessment of the quality of a network topology generated from perturbation experiments. It allows screening of different topologies due to its low demand in experimental data, before further steps (e.g. parameter estimation) are undertaken. Besides highlighting which parts of the topology agree well with existing experimental data, the model also provides predictions that can aid in the design of new experiments.
Finally, we propose that our method can be used to assess the completeness grade of any network. This could be especially useful for GRNs involved in human diseases, where the amount of connectivity is only unknown and/or many genes/interactions are missing.
Methods
Kinetics of the Endomesoderm Network Model
We set k_{ deg }= 0.3/hpf (hours post fertilization), and k_{ trans }= 2/hpf. The values for complex association/dissociation were sampled along with the parameters regulating transcription.
where kdeg·[x] is a degradation term. since we require S_{1} ∈ (0, 1), only one of the two other terms is not equal 0. Thus, by changing Θ and S_{1}, an external input is activated at time point Θ (in the case that S_{1} = 1) or inactivated (when S_{1} = 0). By changing S_{1} and Θ using events, we can exactly control the activity of the external inputs. The exact numerical values for each of the inputs are given in the SBML Model provided as Additional File 5.
For this analysis, the ODE model was simulated using the simulation tool PyBioS [33, 34].
Detection of Topological Perturbation Effects in ODE Models
TPEs of a model are detected by comparison of different simulation results. To this end, we start by simulating the basic (unperturbed) model. Then we simulate a model perturbed in the expression of one gene, by either KD or MOE, and compare the results. Given an initial ODE model, M_{0}, the mRNA concentration of a specific mRNA k is defined as s(k, 0, t). Knockdown models are created by setting the rate law for the mRNA production in question, v_{ transcription }(t) to v_{ transcription }(t)·0.05; over expression is realized by setting v_{ transcription }(t) to v_{ transcription }(t) + 2. For a set of n perturbations, we get models M_{ i }, i = 0...n, where M_{0} denotes the initial, unperturbed model.
A species concentration depends on the initial conditions and parameter sets used. Since the analysis will use multiple parameter sets, we will use s(k, i, j, t) to denote the concentration of species k at time point t using model i and parameter set j.
By choosing a threshold th, we can discriminate between significant and insignificant changes by requiring that for an effect to be significant.
As stated before, s(k, i, j, t) depends on the chosen set of parameters. Consequently, r(k, i, j, t) also depends on the chosen parameter values. In order to find perturbation effects of the model invariant to parameter changes we need to find r(k, i, j, t) that are significant for arbitrary parameter sets.
where ht is a predefined threshold. If none of the conditions in Equation 15 evaluates to true, no topological perturbation effect is defined for the given combination of k, i, t under parameter set J.
Comparison of Topological Perturbation Effects in the Endomesoderm ODE Model to Experimental Data
We used the algorithm described above to detect TPEs of the ODE model constructed from the Endomesoderm Network. We sampled 800 parameter sets from a lognormal distribution with σ = 1.5, μ = 0.5. This distribution was in part chosen to avoid too extreme parameter values that could impede the numerical stability of the ODE system. In this study, we set the thresholds th_{1} = 1, th_{2} = 1.25 and ht = 0.9. As an example, the effect of Pmar1-KD and -MOE on the expression of HesC and Alx1 is shown in a scatter plot in Figure 7 for visual orientation.
In order to compare the detected effects to experimental data, we computed the TPEs for a set of time points T = (14, 19, 25, 33, 45, 66) reflecting the time intervals used under experimental conditions [24]. TPEs have been computed for the endoderm, mesoderm and PMC parts of the model as well as the total model. These TPEs differ because of the different initial conditions used and the different external inputs. The TPEs detected are of qualitative nature. Expression of a gene is unaffected, increased or decreased by a perturbation. The experimental data is quantitative and often ambiguous. Multiple measurements for one gene, perturbation and time interval are listed. In some cases, up- as well as downregulation was detected at one data point. In order to compare the two sets, we discretized the experimental data. We chose to use only unambiguous data, i.e. data points that were affected in the same way (upregulated, downregulated or unaffected) in all measurements.
The experimental data were determined for the whole embryo. Simulation results were computed for cells of different territories. Combination of these territories to a total simulation result is impossible due to the omitted intercellular signaling. Even a combination of the three different territories could deviate from the experimental result because the embryo is made up of more different cells than just endoderm, mesoderm and PMC. To account for these discrepancies between experimental results and simulation in the comparison of the two sets, we define a match between detected TPEs and experimental results as a TPE detected in at least one of the simulated territories that is equal to the experimentally determined effect.
Randomization of Network Topology
To generate randomized versions of the model, we applied a method similar to the switching algorithm used in Milo et al. [27] except for not excluding self-edges since the Endomesoderm network itself already contains self-edges. We randomized the ODE model using the mathematical model provided by PyBioS [33, 34]. The method chooses two nodes in the network, say genes g_{ i }and g_{ j }, at random. A regulatory interaction r(x, y) in the network is defined by its origin x and target y. For g_{ i }and g_{ j }, two edges, say r_{1}(x, i) and r_{2}(y, j), that target g_{ i }and g_{ j }, respectively, are selected at random. The two targets are switched so that r_{1}(x, i) → (x, j) and r_{2}(y, j) → (y, i). This switch is repeated a number of times, usually about 100·N, where N is the number of nodes in the network [27]. For the Endomesoderm ODE model, each switch is realized consistently in each territory. Note that the borders between the three territories and the external inputs are unaffected by this randomization.
Applying this randomization algorithm, general topological features of the network like the number of nodes and edges, the average node degree and the degree distribution are preserved while the individual wirings are changed. After a randomized network has been constructed, TPEs can be computed and compared with experimental data. In this analysis, we constructed 2 randomized models using 3000 randomization steps and simulated each with 100 different parameter sets. To infer the effect of stronger randomization, we also constructed one model using 30000 randomization steps and simulated it with 20 parameter sets.
Availability and requirements
The model can be downloaded from the PyBioS website http://pybios.molgen.mpg.de/Pybios/EndoMeso. An SBML version of the model was submitted to the Biomodels database (MODEL2133240427).
Declarations
Acknowledgements
We thank Andrew Hufton for text corrections. This work was supported by the Max-Planck Gesellschaft zur Förderung der Wissenschaften e.V. and the European Union by grant LHSG – CT – 2004 – 512092. Clemens Kühn is funded by the Deutsche Forschungsgemeinschaft via the International Research Training Group "Genomics and Systems Biology of Molecular Networks".
Authors’ Affiliations
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