Supplementary Information
Transcrição
Supplementary Information
Supplementary Information Mae Woods, Miriam Leon, Ruben Perez and Chris P. Barnes Contents 1 Supplementary Methods 1 1.1 Bayesian inference and model selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Approximate Bayesian Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.1 Parameter inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.2 Model selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.3 Variable selection ABC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.4 Kernels for the indicator variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Functional regions, Bayesian integrals and occams razor . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Biochemical modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.5 Precomputing the prior over networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5.1 General three node network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5.2 General four node network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2 Supplementary Figures 1 1.1 10 Supplementary Methods Bayesian inference and model selection Let θ ∈ Θ be a parameter vector with prior π(θ) and f (y|θ) be the likelihood of the data y ∈ D. In Bayesian inference we are interested in the posterior density f (y|θ)π(θ) . f (y|θ)π(θ)dθ Θ π(θ|y) = R Comparison of a discrete set of models can be be performed using the marginal posterior. Suppose we have a set of competing models M ∈ M = {M1 , M2 ..Mq }. Consider the joint space defined by (M, θ) ∈ M × ΘM ; Bayes theorem can then be written π(M |y) = R f (y|M )π(M ) f (y|M )π(M ) =P , 0 0 0 )π(M 0 )dM 0 f (y|M M f (y|M )π(M ) M 1 where f (y|M ), the marginal likelihood, can be written Z f (y|M ) = π(θ|M )f (y|θ, M )dθ. ΘM Therefore the posterior probability of a model is given by the normalized marginal likelihood which may or may not be weighted depending on whether the prior over models is informative or uniform respectively. 1.2 1.2.1 Approximate Bayesian Computation Parameter inference Consider the case where we cannot write down the likelihood in closed form but we can simulate from the data generating model. We can proceed by first sampling a parameter vector from the prior, θ∗ ∼ π(θ), and then sampling a data vector, x∗ , from the model conditional on θ∗ , ie x∗ ∼ f (x|θ∗ ). This alone gives the joint density π(θ, x). To obtain samples from the posterior distribution we must condition on the data y and this is done via an indicator function, i.e. π(θ)f (x|θ)IAy (x) , π(θ, x|y) = R π(θ)f (x|θ)dxdθ Ay ×Θ where IB (z) denotes the indicator function and is equal to 1 for z ∈ B. Here Ay = {x ∈ D : x = y}, so the indicator is equal to one when the simulated data and the observed data are identical. This forms a rejection algorithm, and in this instance the accepted θ∗ are from the true posterior density π(θ|y). For most models it is impossible to achieve simulations with outputs in the subset Ay and so an approximation must be made. This is the basis for ABC. In the first instance we can replace Ay by Ay, = {x ∈ D : ρ(x, y) ≤ } where ρ : D × D → R+ is a distance function comparing the simulated data to the observed data. We then have π(θ)f (x|θ)IAy, (x) , π(θ)f (x|θ)dxdθ Ay, ×Θ π (θ, x|y) = R where π is an approximation to the true posterior distribution. The rationale behind ABC is that if is small then the resulting approximate posterior, π , is close to the true posterior. We often write the marginal posterior distribution as π(θ|ρ(x∗ , y) ≤ ). Often, for complex stochastic systems, the subset Ay, is still too restrictive. In these cases we can resort to comparisons of summary statistics. We now specify the subset Ay,η, = {x ∈ D : ρS (x, y) ≤ } where η : D → S is a summary statistic and the distance function now takes the form ρS : S × S → R+ . The simplest algorithm is known as the ABC rejection algorithm [1] and proceeds as follows R1 R2 R3 R4 Sample θ∗ from π(θ). Simulate a dataset x∗ from f (x|θ∗ ). If ρ(x∗ , y) ≤ accept θ∗ , otherwise reject. Return to R1. This gives draws from π but can be very inefficient in high dimensional models or when the overlap between the prior and posterior distributions is small. One way to improve the efficiency of the rejection algorithm is to perform sequential importance sampling (SIS) [2]. In SIS, instead of sampling directly from the posterior distribution, sampling proceeds via a series of intermediate distributions. The importance distribution at each stage is constructed from a perturbed version of the previous population. This approach can be used in ABC and the resultant algorithm is known as ABC SMC [3]. Described here is a slightly modified version that automatically calculates the schedule and as such, only the final value, T , needs be specified. To obtain N samples {θ1 , θ2 , θ3 ...., θN } (known as particles) from the posterior, defined as, π(θ|ρ(x∗ , y) ≤ T ), proceed as follows 2 S1 S2.0 S2.1 S2.2 Initialize = ∞ Set the population indicator t = 0 Set the particle indicator i = 1 If t = 0, sample θ∗∗ independently from π(θ) i If t > 0, sample θ∗ from the previous population {θt−1 } with weights wt−1 . ∗∗ ∗ Perturb the particle, θ ∼ Kt (θ|θ ) where Kt is the perturbation kernel. If π(θ∗∗ ) = 0, return to S2.1 Simulate a candidate dataset x∗ ∼ f (x|θ∗∗ ). If ρ(x∗ , y) > return to S2.1 Set θti(= θ∗∗ and dit = ρ(x∗ , y), calculate the weight as 1 if t = 0 wti = π(θti ) PN if t > 0 wj K (θ i |θ j ) j=1 S3.0 t−1 t t t−1 If i < N , set i = i + 1, go to S2.1 Normalize the weights. Determine such that P r(dt ≤ ) = α. If > T , set t = t + 1, go to S2.0. Here Kt (θ|θ∗ ) is the component-wise random walk perturbation kernel that, in this study, takes the form Kt (θ∗ |θ) = θ + U (−δ, δ) where δ = 12 range{θt−1 }. The denominator in the weight calculation can be seen as the probability of observing the current particle given the previous population. The value α is the quantile of the distance distribution from which to choose the next . This is set conservatively (α = 0.3) since taking small steps in the distance schedule can lead to the algorithm getting stuck in local minima [4]. 1.2.2 Model selection Model selection can be incorporated into the ABC framework by introducing the model indicator M and proceeding with inference on the joint space. For example, the ABC rejection algorithm with model selection [5] proceeds as follows MR1 MR2 MR3 MR4 MR5 Sample M ∗ from π(M ). Sample θ∗ from π(θ|M ∗ ). Simulate a dataset x∗ from f (x|θ∗ , M ∗ ). If ρ(x∗ , y) ≤ accept (M ∗ , θ∗ ), otherwise reject. Return to R1. Once N samples have been accepted an approximation to the marginal posterior probability, P (M = m|y), is given by #accepted m . N P (M = m|y) = Model selection can also be incorporated into the ABC SMC algorithm described above [3, 6]. It has been noted that using summary statistics for model selection can be problematic due to information loss [7, 8]. This can give rise to a discrepancy between the Bayes factor calculated from the full data and the summary statistics, unless the summaries are sufficient for the joint space, {M, θ}. We construct the problem in terms of summary statistics of the time course data and define the posterior in terms of the summaries. How model selection on the summaries maps to model selection on any individual time course is irrelevant here. 1.2.3 Variable selection ABC The algorithm presented here makes a simplification to allow for the traversal of large model spaces. We assume that all models are nested within one larger model and that the candidate models share the same parameter 3 space. In terms of model selection we are making the following approximation of the joint space π(θ, M ) = π(θ|M )π(M ) ≈ π(θ)π(M ). This allows for the separation of model and parameter space and transforms the model selection problem into a variable selection one. The main disadvantages of this approach is that model specific parameter spaces (and also priors and perturbation kernels) cannot be constructed. In order to implement this approach we introduce nI model indicators Ik , k ∈ {1, 2, ..nI } which are integer valued parameters Ik ∈ Z. Here we will assume that each indicator takes three values, Z = {−1, 0, +1} (representing repression, missing and activating respectively), although this is not a requirement and could be made more general. Each Ik has a prior specified by a discrete uniform distribution (no particular interaction is favoured) and a perturbation kernel, Kt,I (Ik∗ |Ik ) specified by a transition matrix (see next subsection). Particular models are specified by combinations of the indicator variables. Let I = (I1 , I2 , ...Ik )T be the vector of indicator variables and z = (z1 , z2 ..zk )T be the vector of their integer values corresponding to a particular network structure. We can obtain the posterior probability of a particular network, P r(z), by summing the importance weights for all those particles with indicators I1 = z1 , I2 = z2 , ....InI = znI , or I = z. Therefore we can write P r(z) = N X wi (I1i = z1 , I2i = z2 , ..., Ini I = znI , θi ) i=1 = N X wi (I i , θi )Iz (I i ), i=1 where I denotes the indicator function. This allows the algorithm to explore large numbers of nested models. For example if k = 6 and Z = {−1, 0, +1} the algorithm searches 36 = 729 nested models. If k = 9 and Z = {−1, 0, +1} the algorithm searches 39 = 19683 nested models. The modified algorithm for large model space traversal is given below. To obtain N samples {(I, θ)1 , (I, θ)2 , (I, θ)3 ...., (I, θ)N } from the posterior, defined as, π(I, θ|ρ(x∗ , y) ≤ T ), proceed as follows VS1 VS2.0 VS2.1 VS2.2 Initialize = ∞ Set the population indicator t = 0 Set the particle indicator i = 1 If t = 0, sample (I ∗∗ , θ∗∗ ) from the prior π(I, θ) = π(I)π(θ). If t > 0, sample (I ∗ , θ∗ ) from the previous population {(I, θ)it−1 } with weights wt−1 . Perturb the particle, (I ∗∗ , θ∗∗ ) ∼ Kt,θ (θ|θ∗ )Kt,I (I|I ∗ ) where Kt and Kt,I are the perturbation kernels. If π(I ∗∗ , θ∗∗ ) = 0, return to MS2.1 Simulate a candidate dataset x∗ ∼ f (x|I ∗∗ , θ∗∗ ). If ρ(x∗ , y) > return to MS2.1 Set (I,(θ)it = (I ∗∗ , θ∗∗ ) and dit = ρ(x∗ , y), calculate the weight as 1 if t = 0 wti = π(θti )π(Iti ) PN if t > 0 wj K (θ i |θ j )K (I i |I j ) j=1 VS3.0 VS END t−1 t t t−1 t,I t t−1 If i < N , set i = i + 1, go to LMS2.1 Normalize the weights. Determine such that P r(dt ≤ ) = α. If > T , set t = t + 1, go to LMS2.0. Denote the final population by T Obtain the marginal model probabilities given by PN P r(z) = i=1 wTi (ITi , θTi )Iz (ITi ) 4 Finally we can also define the inclusion probability of any single edge within the network by P r(Ik = zk |.) = N X wi (I i , θi )Izk (Iki ). i=1 1.2.4 Kernels for the indicator variables In principle the kernels for the indicator variables can be specified in be defined through a transition matrix of the form 1 − p p/2 P r(Ik∗ = j|Ik = i) = p/2 1 − p p/2 p/2 number of ways. A dependent kernel can p/2 p/2 , 1−p where p is constant fixed heuristically to give good mixing properties. In the original ABC SMC algorithm general a value p = 0.7 was used. We also defined an adaptive independent kernel such that the perturbation uses information from the previous population. A simple adaptive kernel is given by the transition matrix p−1 p0 p1 P r(Ik∗ = j|Ik = i) = p−1 p0 p1 . p−1 p0 p1 The values within the transition matrix are given by p−1 ∝ P r(Ik = −1) + c/3 p0 ∝ P r(Ik = 0) + c/3 p1 ∝ P r(Ik = −1) + c/3 where the empirical values, P r(Ik = −1), P r(Ik = 0), P r(Ik = −1) are calculated from the previous population. The value c is a regularisation constant and effectively enforces the kernel to be a mixture of the empirical probability distribution and a uniform distribution, which ensures that while the algorithm adapts to the models with highest posterior probability there is always a probability of switching out of the current model. In the main study this kernel is used with c = 1, which is very conservative, and ensures good mixing properties. There are alternative ways to formulate dependent and adaptive kernels using multinomial regression [9], however since here we are dealing with a small number of indicator variables these methods were explored but deemed unnecessary. 1.3 Functional regions, Bayesian integrals and occams razor To more clearly explain the relationship between robustness, model evidence (marginal likelihood) and how complexity is automatically accounted for, it is instructive to a very simple toy scenario depicted in Figure 1. We assume that the prior, P, is uniformly distributed and is depicted by the blue regions in Supplementary Figure 1. We also assume that a small fraction of parameter space is functional (depicted in red in Supplementary Figure 1) ; that is if the parameters lie within this region then the system exhibits the desired dynamical behaviour (oscillations in this case). Within the functional region, F, the likelihood f (O|θ) is equal to one, otherwise it is zero f (O|θ) = 1 if θ ∈ F = 0 otherwise. 5 A B θ2 θ1 b a b b θ1 θ3 θ2 Figure 1: . Examples of two (A) and three (B) dimensional parameter spaces. The prior is depicted by the blue regions and the functional region is depicted in red In addition we can define notation for the size of the prior and functional regions as |P| and |F|. With these simplifying assumptions. the model evidence or robustness in the two dimensional case is given by Z p(O) = f (O|θ)p(θ)dθ Z = f (O|θ)p(θ1 )p(θ2 )dθ1 dθ2 Z 1 dθ1 dθ2 = 2 b F |F| = , |P| which is simply the ratio of the functional region to the prior region. When examining two systems of the same complexity (equal number of parameters) and with identical prior distributions, then the Bayes factor is simply the ratio of the sizes of the functional regions R f (O|θ, M1 )p(θ, M1 )dθ BF12 = R f (O|θ, M2 )p(θ, M2 )dθ |F1 | |F2 | = / |P1 | |P2 | |F1 | = . |F2 | which make sense intuitively. It is straightforward to see that these relationships hold in any number of dimensions. Now imagine the case where a parameter is added and we transition from R2 to R3 space. The functional region remains invariant in the θ1 , θ2 plane and extends an amount a in the θ3 direction (Supplementary Figure 1B). 6 The marginal likelihood becomes Z p(O) = = f (O|θ)p(θ1 )p(θ2 )p(θ3 )dθ1 dθ2 dθ3 |F 2 | a , |P 2 | b where we have used the notation F 2 and P 2 to label the two dimensional region (θ1 , θ2 plane) of F and P respectively. Interestingly, if θ3 is completely uninformative, that is a = b, then the robustness is identical to the two dimensional case. This is also intuitive since adding a parameter that does nothing should not change the calculated robustness. This does highlight a possible issue; models with uninformative parameters are not penalised since they all receive the same robustness. This might be problematic in network inference since one could end up with networks with spurious edges. However in our case, this is not a problem since an edge constitutes at least two parameters (Ik , θ). The only way an edge could be added spuriously is if the dynamical parameter(s) are exactly equal to zero, θ = 0. This has a zero probability of occurring in this analysis. Now consider the comparison of a three and two parameter valued model. The Bayes factor is given by |F 3 | |F 2 | / |P 3 | |P 2 | |F 3 | |P 3 | = 2 / 2 . |F | |P | BF = In order for there to be an increase in robustness, BF > 1 and the functional region must increase by a proportion greater than the proportional increase in the prior region. In the simple case here, adding a parameter implies a penalty in the robustness of |P (n+1) |/|P n | = b, the size of the prior range. This is how the penalty for model complexity, sometimes called Occam’s razor or equivalent to “the simplest model is the best”, is automatically included in our definition of robustness. The results gained from this informal examination of the simple case generalises to higher dimensions, non-uniform priors and more complex likelihood functions. 1.4 Biochemical modelling An example of how the promoters are modelled in this study is given in Supplementary Figure 2. In this case, the promoter is under the regulation of both a repressor and an activator, and thus has two operator sites. Throughout this analysis, all repressors and promoters are assumed to form dimers. The relative terms are constructed using mass-action type arguments and the amount of the dimer is assumed to be proportional to the square of the amount of the monomer, as described in the Materials and Methods section of the main text. The probability of transcription, ptransc , is calculated from the ratio of states in which RNAP is bound, to all possible states (the partition function). We do not assume a maximum limit on the number of regulators, and have up to four (including auto-regulation) in the four gene network, although this could in principle be restricted to a lower number. 7 Terms DNA O1 O2 X 1 O2 X k1 O2 X k2 R2 = X k3 A2 = Promoter RN AP O1 R2 O1 A2 RN AP O1 O2 ptransc = k2 2 R Kd = δR2 k3 2 A Kd = σA2 k1 + σA2 1 + k1 + δR2 + σA2 Figure 2: An example model of a promoter under the regulation of a repressor and an activator. 8 1.5 Precomputing the prior over networks In this section we outline how the prior over model space was calculated. Essentially the entire space of models is enumerated and then symmetrical or otherwise uninteresting networks are excluded, which is equivalent to setting the prior probability to zero. In principle, rather than setting the prior probability of topologies to zero, one could weight topologies according to other criteria. The adjacency matrices for the general three and four node networks I1 I8 I1 I8 I4 I5 I2 A3n = I5 I2 I9 A4n = I7 I6 I7 I6 I3 I14 I15 are given by I4 I11 I9 I12 . I3 I13 I16 I10 Two networks G1 and G2 , with associated adjacency matrices A1 and A2 are isomorphic if there exists a permutation matrix P such that A1 = P A2 P −1 . Additionally permutation matrices are orthogonal so P −1 = P T . 1.5.1 General three node network The case of the general three node network can be handled analytically. The output node, C, is fixed and we need only concern ourselves with the symmetry in nodes A and B. We introduce the permutation matrix PAB 0 1 0 PAB = 1 0 0 0 0 1 T but in general P 6= P T . We divide the possible networks into two types; Note that in this case PAB = PAB those that are invariant under the transformation and those that aren’t. The invariant topologies are obtained by solving A0 = PAB APAB . Writing out A explicitly in terms of its elements and solving gives the following requirements on the invariant topologies I1 A I8 a011 = a22 a012 = a21 a013 = a23 B a031 = a32 I5 I7 I6 I4 C I9 I2 I3 which gives a total of 35 = 243 topologies. Assuming only half of the remainder are necessary gives a total of 243 + 9720 = 9963 topologies. The breakdown in terms of nodes and edges are given below. nodes number 1 135 9 2 216 3 9612 edges number 1.5.2 0 1 1 10 2 76 3 344 4 1020 5 2040 6 2704 7 2336 8 1160 9 272 General four node network The four node network case is difficult to handle analytically since there are now three symmetry operations which do not commute. Assuming that node D is fixed we can form the following three permutation matrices 1 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 0 0 1 0 0 0 PAB = 0 0 1 0 , PAC = 1 0 0 0 , PBC = 0 1 0 0 . 0 0 0 1 0 0 0 1 0 0 0 1 There are in total 316 possible topologies. For each topology, we computationally applied the three symmetry operations A0 = PAB APAB A0 = PAC APAC A0 = PBC APBC . The transformed networks were stored and then duplicate topologies removed, leaving a total of 7886403 topologies (around 18%). In this case, to reduced the model space further we also removed 294300 topologies in which node D was unconnected to the rest of the network, giving a total of 7592103 topologies (a directed connection from node A, B or C). The breakdown in terms of nodes and edges are given below. nodes number edges number edges number 9 998704 0 0 1 2 2 59 10 1422060 1 0 3 702 2 2754 4 4842 11 1578856 3 29889 5 23230 12 1341760 4 7559460 6 84694 7 242874 8 552960 13 846176 14 375488 15 105264 16 14432 The precomputation of the prior was done in Python using the SciPy package. We expect there to be further symmetries within the resultant topologies, which could be problematic if we were considering individual networks. However since in the main analyses we calculate posterior probabilities over collections of topologies this is not an issue. 2 Supplementary Figures 10 k1_1 k1_2 R1 R2 s_1s s_2t1 d_2s d_1t1 tlA tlB dA dB dmA dmB k1_1 k1_2 R1 R2 s_2s s_1t1 d_1s d_2t1 tlA tlB dA dB dmA dmB Figure 3: Two node oscillators: (top) full posterior distributions for model M1, (bottom) full posterior distribution for model M2 11 M1 sde 800 M2 sde 600 Freq Freq 600 400 400 200 200 0 0 mA A mB B mA A mB B species species M1 sde M2 sde 8e−04 4e−04 density density 6e−04 4e−04 3e−04 2e−04 2e−04 1e−04 0e+00 0e+00 0 2500 5000 7500 10000 0 system_size 2500 5000 7500 10000 system_size Figure 4: Two node oscillators: M1 and M2 comparisons. (Top) Relative frequencies of the species with the highest average amount over 1024 simulations from the posterior. (Bottom) Average system size over 1024 simulations from the posterior. 12 Posterior probability Network 0.8076 0.0855 0.0458 0.0331 A A A A B Edges 1 -1 -1 1 B -1 1 1 -1 B -1 0 -1 -1 B -1 -1 0 1 Figure 5: Robustness of two node oscillators under the objective of regular oscillations (S2). 13 1,0,0 1,1,0 1,0,−1 1,0,1 ● ● ● ● ● ● ● ● ● ● ● −1,1,1 −1,1,−1 ● ● ● ● ● ● −1,−1,−1 −1,−1,1 ● ● −1,0,−1−1,−1,0 0,1,1 0,1,−1 −1,1,0 −1,0,1 0,−1,1 ● ● 0,0,1 1,−1,1 1,−1,−1 0,0,−1 0,−1,0 −1,0,0 1,1,1 1,1,−1 0,1,0 ● 1,−1,0 0,0,0 ● Figure 6: Graph of auto-regulation for robust ring oscillators. The nodes correspond to different configurations of auto-regulation on nodes A,B,C respectively. For example −1, 0, 1 indicates negative auto-regulation on A, no auto-regulation of B and positive auto-regulation on C. The size of the nodes is proportional to the robustness of the resultant ring oscillator (averaged over all topologies containing the auto-regulatory motif). 14 R1 R2 R3 d_3t tlA tlB tlC dA dB dC dmA dmB dmC R1 R2 R3 d_3t tlA tlB tlC dA dB dC dmA dmB dmC Figure 7: Posterior distributions of the production and decay rates for the ring oscillator (top) and the ring oscillator with positive auto-regulation on A (bottom). In the former the decay rates of protein C and all mRNA species are constrained to be high. In the latter only the decay rates for C (protein and mRNA) are constrained to be high. 15 20 ● ● ● 15 Bayes factor ● ● ● ● 10 ● ● ● ● 5 ● ● ● ● ● 0 19.97 C B 9.56 C B C C C B C C C B C C B 0.39 C C C 0.23 C B C C B C B B C C B A C C A C B 0.2 A C B 0.25 A B 0.21 A 0.4 A B 0.27 A 0.22 A 0.49 A 0.3 A B A 0.6 A B A B 5.97 A 0.73 A B 7.99 A B 5.99 A 11.23 A B 8.04 A B 6.43 A 14.57 A B 8.67 A B 6.48 15.03 A A B C Figure 8: Robustness of ring oscillators under the objective of regular oscillations (objective S2). (Top) Distribution of Bayes factors. (Bottom) Most robust 12 networks (left) and top 12 interactions ranked by inclusion probability (right). 16 −1.0 Pearson correlation: 0.807 ● −1.5 ●● −2.0 ● ● ● ● ● ● ● −2.5 ● −3.0 −3.5 ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●●● ● ● ●● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ●●●● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● −4.0 log10(relative robustness) s2 ● ●● ●● ● ● ● ● ● ● ● ● ● −4.5 ● ● −4.0 −3.5 −3.0 −2.5 −2.0 −1.5 log10(relative robustness) s1 dA dA dB dB dC dC dmA dmA dmB dmB dmC dmC Figure 9: Comparison of models under objective S1 (fixed frequency) and objective S2 (regular oscillations). (Top) Comparisons of the relative robustness of models under both objectives. (Bottom) Posteriors of ring oscillators under both objectives S1 (left) and S2 (right). 17 1000 1000 2000 3000 750 750 1500 750 1000 1000 0 0 3 4 5 6 7 8 9 500 250 500 250 0 3 regulatory interactions 4 5 6 7 8 9 Freq Freq 500 Freq Freq Freq 2000 500 250 0 3 regulatory interactions 4 5 6 7 8 9 0 3 regulatory interactions 4 5 6 7 8 9 3 regulatory interactions 4 5 6 7 8 9 regulatory interactions 800 200 600 200 200 0 100 4 5 6 7 8 regulatory interactions 9 0 3 4 5 6 7 8 9 4 5 6 7 8 200 100 0 3 regulatory interactions 60 30 50 0 3 Freq 400 Freq 400 Freq 150 Freq Freq 300 90 600 0 9 3 regulatory interactions 4 5 6 7 8 regulatory interactions 9 3 4 5 6 7 8 9 regulatory interactions 1.00 0.75 fraction regulation neg mix 0.50 pos none 0.25 0.00 1 2 3 4 5 6 7 8 9 10 motif Figure 10: Robust three node oscillators. (Top) Total number of regulatory interactions within the top ten categories. (Bottom) The fraction of networks containing positive, negative and mixed auto-regulatory feedback loops. 18 0 20 40 0.98 ● 20 ● ● ● ●● ● ● ●● ● ●● ● ●● ●●● ●● ●● ●● ●● ●● ●● ● ●● ● ● ● ● ● ● ● ● ● ●● ●● ●● ●● ●●● ●● ● ● ● ●● ● ● ●● ●●● ●● ●● ● ●● ●●● ● ● ●● ●● ●● ●● ●●●● ●● ● 0 0 ● ●● ●● ● ●● ● ●● ●●● ● ●● ●●● ●● ● ●● ●● ● ● ●● ●● ● ●● ●●● ●● ● ● ●●● ● ● ●● ●●● ● ●● ● ●●●● ● ● ● ● ●● ● ● ● ● freq/100 (resim) ● 20 ● ●● ●● ● ●● ●● ● ●● ●● ● ●● ● ● ●● ● ●●● ●● ●● ● ●● ●● ● ●●● ●● ● ●●● ●● ●● ●● ●● ●●●● ●● ●● ●● ● ● ● ● ● ● ●● ●● ●●●● ●● ● ●● ●● ●● ●● ●●● ● ● ●●● ● ● ● ● 40 0 freq/100 (orig) ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● 2 3 4 4 3 2 2 2 2 3 4 log10(amplitude) (orig) 4 2 5 4 3 2 5 4 2 3 4 log10(amplitude) (orig) 3 3 2 5 0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● 2 3 ● ● 4 2 3 4 log10(amplitude) (orig) 3 2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● 2 3 4 5 5 4 2 5 ● ● ● ● log10(amplitude) (orig) 0.997 3 40 0.997 4 5 5 ●● ● ● ● ● ●● ● ●● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● 20 freq/100 (orig) log10(amplitude) (orig) 0.997 4 freq/100 (resim) freq/100 (resim) 2 5 5 ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ●● ● ● ●●● ●● ●● ●● ● ●●● ●● ●● ●● ●● ●●●●● ●● ● ● ●● ●● ●● ●● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●●● ●● ● ●● ● ●● ● ●●● ● ● ● ● ● ● ● ● 5 0.999 4 ● ● ● 20 40 5 3 ● freq/100 (orig) ● 40 0.919 40 0 20 20 freq/100 (orig) ● ● ● ●● ● ●● ● ● ●● ●● ●● ●● ●● ●●● ●● ● ● ●● ●● ●● ● ● ●● ● ● ● log10(amplitude) (orig) 0.998 0 20 0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 2 5 ● log10(amplitude) (resim) log10(amplitude) (resim) 3 3 3 0 40 0.903 40 40 0.998 4 20 0 20 5 ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 20 ● ●● ●●●●● ● ● ● ●● ●● ●● ● ●● ●●● ●● ●● ●●●● ● ● ●● ● ● ●● ● ● ● ● ● ●● ●● ● ●● ● ●● ●● ● ● ● ●● ● ● ●● ● ●● ● ●●●●● ● ●● ● ● ● ● ●● ● ● ●●●●● ● ● ● ● ●● ●● ●● ●● ● ● ● ● ●● ●● ●●● ●● ● ●● ● ●● ● ●●● ● ● ● ● ● ●● ●● ●● ● ●● ●●● ● ●● ●● ●● ●● ●● ●● ● ● ●● ● ● ● ●●● ●●● ● ● ● ●● ● ● ● 0.984 40 freq/100 (orig) freq/100 (orig) log10(amplitude) (orig) ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● 0 ● ●● ●● ●● ●● ●● ●● ●● ● ● ● ●● ●● ●● ●● ●●● ●● ● ● ● ● ● ● ● ●● ●● ● ●● ●● ●● ●● ● ● ● ● ● ● 0 ● 5 0.999 4 40 0.998 log10(amplitude) (orig) 5 20 0 20 log10(amplitude) (resim) log10(amplitude) (resim) log10(amplitude) (resim) 2 ● ● 5 0.996 3 ● 0 40 0.905 40 freq/100 (orig) 5 4 ●● ● ● 0 20 ● ● ●● ●● ●● ●● ● ● ●● ●●● ● ●● ●● ● ● ● ●● ●● ●● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ●● ● 0.977 40 20 freq/100 (orig) log10(amplitude) (resim) freq/100 (resim) ● 40 0 freq/100 (orig) 20 freq/100 (resim) 0 0.978 40 log10(amplitude) (resim) 40 ● ●●● ● ●● ● ● ●●●● ●● ●●●● ●●● ● ● ●●●● ●● ●●●● ● ● ● ● ● ●●●● ● ● ● ● ●●● ● ● ● ●● ● ●● ●● ●● ● ● ● ● ● ● ●●● ● ● ● ●● ● ● ● ● ●● ●● ● ● ● ●● ● ●● ●● ● ● ●●● ●● ●● ● ● ●● ● ● ●● ●● ●● ●● ●● ●●●● ● ● ● ● ● ● ●● ●● ●● ●● ●●● ● ●● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ●●● ● ●● ●●● ●● ●● ● ● ● ● ● ● ● ● ● log10(amplitude) (resim) 20 freq/100 (orig) 20 freq/100 (resim) 0 ● 0 ● ● ● ●● ● ● ●● ●● ● ●● ●● ● ● ● ●● ●● ●● ● ●● ●● ●● ● ● ● ● ● ● ● ● ●● ●● ●● ● ● ●● ● ●● ● ● ● ● ● ●● ●● ● ● ● ● ●● ●● ● ●● ●● ●● ●● ● ● ● ●● ● ●● ●● ●● ●● ●● ●●● ● ●● ●● ● ● ● ● ● ● ● ● ● ●● ●● ●● ●● ●● ●●●●●●● ● ● ● ● ● ● ● ● ● ● ●● ●● ●●●●● ●● ● ●● ● ● ● ● 0.986 40 log10(amplitude) (resim) 0 ●● 20 ●● ●● ● ● ● ● ● ●● ●● ● ●● ● ● ● ● ● ●● ● ●● ● ●● ●● ● ●● ● ● ● ●● ●● ● ●● ●● ●● ●● ●●● ● ●● ●● ●● ●● ● ●● ●● ●● ● ● ●● ●● ● ● ●● ● ●● ●● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ●●● ●● ●●● ●● ● ● ● ●● ●●●● ●● ● ● ●● ●● ●●● ●● ●● ●● ●●● ●● ●● ●● ●● ●●●● ●●● ● ● ●● ●●● ●● ●● ●● ●●● ●● ●● ● ● ● ● ●●● ● ● ●●●● ● ● ● ●●● ● ● ● ● ●●●●● ● ●● ● ●● ●●●● ●●● ●● ● ●● ●● ●● ● ● ● ● ●● ●●● ● ● ●● ●●● ●● ● ● ● ●● ● ●● ● ●● ●● ●● ●● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ●●●● ● ●● ● ●● ●● ●● ● ● ●● ●● ● ●● ● ● ●●●● ● ● ●● ● ●● ●●●● ●●● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ●● ●● ●●● ●● ● ●● ●●●● ●● ● ● ●●●● ●● ●● ● ● ●● log10(amplitude) (resim) ● ●●●●●● ● ● ●● ●● ● ●●●● ●● ●● ●● ●● ●● ●● ● ● ● ● ● ● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ●● ●● ● ● ●● ●● ● ●● ● ●● ● ● ● ● ●●●●●●●● ● freq/100 (resim) 20 0.961 40 freq/100 (resim) freq/100 (resim) freq/100 (resim) 0.882 40 ●●●●● ●●●● ● ●● ● ●●● ● ●● ● ● ● ● ●● ● ● ● ●●● ● ●●● ●● ● ●●● ● ● ●● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ●● ● ●● ● ● ● ● ●● 2 3 0.998 4 3 2 4 log10(amplitude) (orig) 5 ●● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● 2 3 ● 4 5 log10(amplitude) (orig) Figure 11: Resampling and resimulation of the the top ten robust three node oscillators. (Top) Frequency. (Bottom) Amplitude. 19 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 50 ● ● frequency/100 40 ● ● ● ● 30 ● ● ● ● 20 ● ● ● ● ● ● ● ● ● 10 0 ● ● ● ● ● ● ● 0:0 0:1 ● ● ● 1:0 1:1 high deg mC,C,mB,B : S9 and S7 activating Figure 12: Boxplot showing the frequency of three node oscillators classified by whether they have low/high degradation rates and whether both I9 and I7 are positive. 0.20 0.4 group group deg ~ U(0,1) 2 node deg ~ U(0,10) ring density density 0.3 0.15 0.2 3 node 0.10 4 node 0.05 0.1 0.0 0.00 0 20 40 0 20 frequency/100 40 frequency/100 1.00 1.00 density density 0.75 0.50 0.25 0.75 0.50 0.25 0.00 0.00 1 2 3 4 5 0 log10(amplitude) 1 2 3 4 5 log10(amplitude) Figure 13: (Left) Density plots of the marginal amplitude and frequency plots for the two, three and four node systems. (Right) Reanalysis of the three gene network with reduced prior range on the degradation rates. 20 1.00 0.20 prior, posterior weight prior, posterior weight 0.75 0.15 variable prior posterior 0.10 variable prior 0.50 posterior 0.25 0.05 0.00 0.00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 2 3 regulatory interactions 4 number of genes ● ● log10(relative robustness) −1.0 ● ● −1.5 −2.0 −2.5 3 4 5 6 7 8 9 10 11 12 core regulatory interactions Figure 14: (Top) Prior and posterior for the number of edges and number of nodes when examining the four-node networks. The relative robustness is given by the ratio of posterior to prior. (Bottom) Oscillator robustness as a function of number of core regulatory interactions (ignoring auto-regulation). 21 f/100, orig vs resim (sde), cor= 0.93 A, orig vs resim (sde), cor= 0.99 40000 30000 Amplitude resim (sde) f/100 resim (sde) 40 count 100 50 20 count 1000 750 20000 500 250 10000 0 0 10 20 30 40 50 0 10000 20000 30000 40000 f/100 original (sde) Amplitude original (sde) f/100, orig vs resim (mjp), cor= 0.82 A, orig vs resim (mjp), cor= 0.98 40000 Amplitude resim (mjp) f/100 resim (mjp) 40 count 80 60 40 20 20 30000 count 1000 750 20000 500 250 10000 0 0 10 20 30 40 50 0 f/100 original (sde) 10000 20000 30000 40000 Amplitude original (mjp) Figure 15: Calculated frequency and amplitude for the ring oscillator after re-simulation assuming a system of stochastic differential equations (sde) and a Markov jump process (mjp). The same parameter sets were used, representing the oscillating region of parameter space for the ring oscillator. The top plots show the correlation after re-simulation using sde. The bottom plots show the correlation after re-simulation using mjp. While re-simulation with mjp shows higher noise in the frequency, the correlation with the original sde output is still very high (0.82 using spearman rank). 22 References [1] J K Pritchard, M T Seielstad, A Perez-Lezaun, and M W Feldman. Population growth of human Y chromosomes: a study of Y chromosome microsatellites. Mol Biol Evol, 16(12):1791–8, Dec 1999. [2] P Del Moral, A Doucet, and A Jasra. Sequential monte carlo samplers. J Roy Stat Soc B, 68:411–436, Jan 2006. [3] Tina Toni, David Welch, Natalja Strelkowa, Andreas Ipsen, and Michael P H Stumpf. Approximate Bayesian computation scheme for parameter inference and model selection in dynamical systems. Journal of the Royal Society Interface, 6(31):187–202, Feb 2009. [4] Daniel Silk, Sarah Filippi, and Michael P H Stumpf. Optimizing threshold-schedules for sequential approximate Bayesian computation: applications to molecular systems. Statistical applications in genetics and molecular biology, 12(5):603–618, October 2013. [5] Aude Grelaud, Christian P Robert, and Jean-Michel Marin. ABC methods for model choice in gibbs random fields. Cr Math, 347(3-4):205–210, Jan 2009. [6] Tina Toni and Michael P H Stumpf. Simulation-based model selection for dynamical systems in systems and population biology. Bioinformatics, 26(1):104–10, Jan 2010. [7] Christian P Robert, Jean-Marie Cornuet, Jean-Michel Marin, and Natesh S Pillai. Lack of confidence in approximate Bayesian computation model choice. Proceedings of the National Academy of Sciences of the United States of America, 108(37):15112–15117, September 2011. [8] Chris P Barnes, Sarah Filippi, Michael P H Stumpf, and Thomas Thorne. Considerate approaches to constructing summary statistics for ABC model selection. Statistics and Computing, 22(6):1181–1197, 2012. [9] Christian Schäfer and Nicolas Chopin. Sequential Monte Carlo on large binary sampling spaces. Statistics and Computing, 23(2):163–184, November 2011. 23
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