Computational analysis of viable parameter regions in models of synthetic biological systems
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Abstract
Background
Gene regulatory networks with different topological and/or dynamical properties might exhibit similar behavior. System that is less perceptive for the perturbations of its internal and external factors should be preferred. Methods for sensitivity and robustness assessment have already been developed and can be roughly divided into local and global approaches. Local methods focus only on the local area around nominal parameter values. This can be problematic when parameters exhibits the desired behavior over a large range of parameter perturbations or when parameter values are unknown. Global methods, on the other hand, investigate the whole space of parameter values and mostly rely on different sampling techniques. This can be computationally inefficient. To address these shortcomings ’glocal’ approaches were developed that apply global and local approaches in an effective and rigorous manner.
Results
Herein, we present a computational approach for ’glocal’ analysis of viable parameter regions in biological models. The methodology is based on the exploration of high-dimensional viable parameter spaces with global and local sampling, clustering and dimensionality reduction techniques. The proposed methodology allows us to efficiently investigate the viable parameter space regions, evaluate the regions which exhibit the largest robustness, and to gather new insights regarding the size and connectivity of the viable parameter regions. We evaluate the proposed methodology on three different synthetic gene regulatory network models, i.e. the repressilator model, the model of the AC-DC circuit and the model of the edge-triggered master-slave D flip-flop.
Conclusions
The proposed methodology provides a rigorous assessment of the shape and size of viable parameter regions based on (1) the mathematical description of the biological system of interest, (2) constraints that define feasible parameter regions and (3) cost function that defines the desired or observed behavior of the system. These insights can be used to assess the robustness of biological systems, even in the case when parameter values are unknown and more importantly, even when there are multiple poorly connected viable parameter regions in the solution space. Moreover, the methodology can be efficiently applied to the analysis of biological systems that exhibit multiple modes of the targeted behavior.
Keywords
Biological model Repressilator AC-DC circuit Biological D flip-flop Computational analysis Genetic algorithms Principal components Viable parameter regions RobustnessAbbreviations
- AC-DC
Alternative current-direct current
- CLK
Clock
- GA
Genetic algorithm
- GRN
Gene regulatory network
- MSE
Mean squared error
- QSSA
Quasi steady-state approximation
- SCSA
Structural and correlative sensitivity analysis
- SSA
Stochastic simulation algorithm
- TF
Transcription factor
Background
Biological oscillators govern various biological processes, such as cellular respiration, cardiac functions, and circadian rhythms [1, 2, 3]. In the terms of synthetic biology, the research of oscillatory systems is motivated by (1) a better understanding of known biological systems [2, 4, 5, 6], and (2) by the development of systems that could potentially be used in practical applications [7, 8]. An implementation of the first synthetic repressilator by Elowitz and Leibler [4] was, together with the synthetic toggle switch by Gardner et al. [9], an important breakthrough in synthetic biology. Since then the focus has shifted from simpler to more complex biological systems [10]. For this reason, the development of robust and fast-response systems is of vital importance. For example, Fink et al. [7] designed an artificial fast-response system in mammalian cells that can respond to chemical signals in minutes rather than hours. In the terms of optimization, this can be translated to multiobjective optimization [11]. Otero-Muras and Banga [12] recently proposed a multiobjective optimization framework for synthetic biology based on the Pareto optimality. This framework is however limited to the library of synthetic parts, which can be a potential limitation. Moreover, when designing complex biological systems, the desired modes of behavior should not be the only criteria. One should also take into an account the system’s robustness, i.e. its stability in terms of correct behavior for a large range of different perturbations of extrinsic and intrinsic factors. If two systems exhibit the same required dynamics, then the more robust system should be preferred [13]. This allows for the development of more efficient and stable biological systems. In order to determine the robustness of the system, one must be able to efficiently explore and characterize its parameter space, for which mathematical modeling is usually applied [14]. To find the optimal parameters that exhibit the desired behavior, different heuristic approaches, such as genetic algorithms (GAs) can be used [14, 15, 16]. While GAs have numerous applications, they usually provide only a single near-optimal solution. However, these approaches do not give us an insight into the shape of the solution space and the robustness of the acquired solution. Other approaches are focused on the efficient investigation of the whole parameter regions for which the system displays some predefined behavior. These regions define so-called viable parameter space. Identification of the viable parameter space allows for a more thorough analysis in the context of system’s robustness, sensitivity and possible modes of behavior [13, 17, 18]. Schillings et al. [18] used adaptive Smolyak interpolation that relies on sparse polynomial approximations to characterize the solution space of biochemical networks. The assumption here is that the function we want to interpolate is sufficiently smooth, which is not always the case. This problem can to some extent be addressed with the adaptive interpolation. Li et al. [19] introduced structural and correlative sensitivity analysis (SCSA) which belongs to the family of global sensitivity methods based on the decomposition of variance. Since the viable parameter spaces only represent a small fraction of all feasible solutions, we are more interested in viable regions, and not on the solution space as a whole. Hafner et al. [13] developed a ’glocal’ robustness analysis and model discrimination method that can be used for the analysis of circadian as well as other oscillators. This approach allows us to efficiently explore the model’s parameter space and assess its robustness. One of its major drawbacks is that it is not applicable to biological systems with high dimensional and poorly connected viable parameter regions. Two regions are poorly connected if one cannot traverse the solution space from one viable region to the other with the arbitrarily small steps, while constantly preserving the viability of the current solution. This is not problematic for the evolutionary developed systems, where the viable solution space is usually connected because natural systems have evolved through small, gradual changes of individual biochemical parameters. And while this may be true for the naturally occurring motifs, it is not necessarily the case for the synthetically developed gene regulatory networks (GRNs). When designing synthetic GRNs, one could choose different parts, e.g., transcription factors (TFs) with similar behavior and different kinetic properties, such as binding-site affinities and degradation rates. This problem was also addressed by Zamora-Sillero et al. in [17], where they proposed an efficient ellipsoid based sampling. The limitation of this approach is that the increasing dimensionality may exponentially increase the number of iterations needed to identify all viable parameter regions. This can occur if the viable solution space is loosely connected.
Herein, we present an improved ’glocal’ approach for the computational analysis of viable parameter spaces in high-dimensional dynamical models of biological systems. The methodology is based on the robustness estimation and model analysis methodology described by Hafner et al. [13] and is due to exhaustive sampling with GAs and clustering not only limited to models with connected solution spaces. The methodology consists of multiple steps, i.e. (1) the estimation of viable parameter regions with GAs, (2) efficient exploration of viable regions with local sampling, and (3) robustness estimation for each of the viable regions. Our methodology differs from the one introduced by Hafner et al. [13] in two main aspects. Firstly, we employ GAs for the initial estimation of viable parameter regions, whereas Hafner et al. relies on the literature available data. The problem is that the number of already published viable parameter values can be quite limited for a particular system, and can guide the exploration in the wrong direction. The second important difference is that our approach can account for and discriminate between multiple poorly connected viable regions, whereas the ’glocal’ methodology by Hafner et al. is limited to a single viable region. Moreover, our methodology can be applied to the systems that exhibit multiple modes of behavior, such as alternative current (AC)-direct current (DC) circuit [20]. We evaluate the proposed methodology on the repressilator model, on the model of the AC-DC circuit that can switch between the oscillatory and bistable behavior as described in [20] and on the model of the biological edge-triggered D flip-flop in a master-slave configuration proposed by Magdevska et al. [14]. We perform the analysis of viable regions of the repressilator model with different cost functions. We analyze four different versions of the D flip-flop model, which differ in the functional forms describing the protein degradation (Michaelian versus linear functions) and transcription factor binding at promoter level (competitive versus independent). We validate the results obtained with deterministic simulations with the additional stochastic simulations. These are performed on the randomly selected samples from the viable regions of each model. The proposed methodology is efficient and thorough, and can be applied for the model-to-model comparison in terms of their robustness. Finally, it is not limited only to systems that exhibit oscillatory dynamics, but can be applied to complex biological systems with arbitrary dynamics.
Methods
Global estimation of viable parameter regions
Possible ranges of parameter values we used in our models
Parameter | Span | Unit |
---|---|---|
Transcription | 10^{−2} - 50 | h^{−1} |
Translation | 10^{−2} - 50 | h^{−1} |
Protein production | 10^{−1} - 50 | h^{−1} |
Protein degradation | 10^{−3} - 50 | h^{−1} |
mRNA degradation | 10^{−1} - 100 | h^{−1} |
Dissociation constant | 10^{−2} - 250 | nM |
Michaelis constant | 10^{−2} - 250 | nM |
Protease concentration | 10 - 1000 | nM |
Dilution rate | 0.6 | h^{−1} |
Hill coefficient | 1 - 5 | - |
Reaction volume (V_{R}·N_{A}) | 1 | nM^{−1} |
where γ_{i} is the i-th peak, P is the number of peaks, and σ_{i} is standard deviation for the i-th peak in the neighboring window of size 3. Ideal sine signals have only one prominent peak. To avoid rewarding the signals with many peaks, we consider only average standard deviation per peak. The minimal size 3 for the window was chosen in order to promote high and narrow peaks. This cost function is defined more loosely in order to not overlook the large range of possible rigid oscillating signals regardless of their amplitude and period, while still favoring undamped oscillations with high amplitudes and a stable periods. We applied both cost functions to the repressilator model to analyze the size and shape of its viable parameter regions.
In the first step of the proposed methodology, the viable regions are estimated by the GA. GAs are inspired by natural evolution and are often used to solve hard optimization problems [15]. Subjects within the population gradually evolve by the means of genetic operators, i.e. mutation, reproduction, and selection. Our initial population consisted of 5000 randomly generated candidates. Each candidate θ was represented as a vector of biochemical parameters mutated with predefined probability. In the literature mutation probabilities most often range on the order of 0.01 per position. While this is much higher than in biology, mutation rates should be chosen on the properties and the difficulty of the problem we are aiming to solve [21]. In our case, every biochemical parameter was multiplied with a random value between 0.8 and 1.2 with the probability 0.75 (high mutation probability was set in order to promote greater exploration of solution space). Every parameter can, therefore, increase or decrease in each iteration of GA. Reproduction was implemented using the two-point crossover. Unlike mutation, which serves as a fine-tuning mechanism, the crossover introduces a certain amount of variability in the population, which makes the problem less susceptible to local extrema. At the end of every iteration of genetic algorithm, subjects are evaluated with the appropriate cost function, and only the fraction of the individuals are chosen for the next generation by tournament selection. We used the tournament size of a tenth of the entire population. In contrary to the traditional use of GAs, we sampled all viable subjects, from which the initial viable set ν^{(0)} is composed. The exploration of solution space stops when the maximal number of generations is reached. To obtain only the approximate estimation of viable regions, the total number of generations should not be too high. We terminated our GA after 10 generations.
Efficient local sampling
where S^{(i)} is a set composed of candidates for viable solutions in i-th iteration of size N (in our case N=10^{5}), E[ν^{(i−1)}] is a mean of the viable candidate solutions in the set ν^{(i−1)} obtained from the previous iteration of the sampling process, ξ_{j} is the j-th Gaussian sample along the principal components of viable set ν^{(i−1)}, and λ^{(i−1)} is the variance scaling factor. Note that the scaled variance should always be greater than the initial variance of principal components in order to cover the whole viable solution space. We set the initial value of λ^{(0)} to 4 and decreased it linearly to 2 in the last, in our case tenth iteration. In this way we focus only on the potential solution areas and avoid unnecessary sampling. At the end of every iteration new viable set ν^{(i)} is obtained by evaluating candidates in set S^{(i)}. For more details please refer to [13].
where k is a number of clusters and A is a constant. If in reality our data consists of K well separated clusters, then the log(W_{k}) will decrease faster than its expected rate E^{∗}[log(W_{k})] for k≤K and slower for k>K. Hence, when k=K the gap will be the largest. Similarly, we set the number of clusters in our data according to the largest gap obtained by gap statistic. In [22], the authors proposed two different choices for the reference distribution, namely (1) generate each reference feature uniformly over the range of the observed values, and (2) generate the reference features from a uniform distribution over a bounding hyper-box B aligned with the principal components of the data. We have chosen the second approach since it is invariant to the rotation of the data. Note that the same technique was used by Hafner et al. [13] for the purposes of Monte Carlo integration. For more information about gap statistic see [22].
When a viable area is recognized as a separate region, it is extensively explored regardless of any other regions by an iterative procedure described in Eq. (5). This, however, poses a threat that explored viable regions overlap, which makes it harder for model-to-model comparison. Alternatively, we can compare only the most robusts regions of both models or combine all regions into a single set.
This allows us to avoid unnecessary clustering at the beginning of the sampling process when the number of viable candidates does not suffice for representative clustering results. This iterative process stops when all regions are sufficiently explored, i.e. principal components do not change over next iterations or the maximal number of iterations is reached.
Robustness analysis
where Φ^{−1} is inverse cumulative distribution function of Z. For a confidence level of 0.95, we get \(|S| \geq \left | \frac {0.98 * \textit {Vol}(B)}{\delta } \right |^{2}\). In other words, if we want to assess the viable volume within one percent of the total bounding box volume with the confidence level of 0.95, S should contain at least 10^{4} samples.
Results
Repressilator
AC-DC circuit exhibits bistability and oscillations
We regarded a parameter point viable if its response exhibited oscillations or bistability. We tested the bistability with two scenarios. First, the initial concentration of protein X was low and Y was high and vice versa in the second scenario. Throughout the simulation, the protein with the initial high concentration should stabilize at 400 nM and the protein with initial low concentration should stabilize at 0 nM. To optimize the bistable behavior we resorted to the cost function (Eq. 3), where we directly compared the response in the time domain instead of in the frequency domain. If the observed response did not deviate, on average, from the ideal signal for more than 4 nM, we considered a parameter point viable.
Edge-triggered d flip-flop in a master-slave configuration
Approximated relative volumes of the feasible solution spaces in the D flip-flop model
Ω_{1} | Ω_{2} | Relative volume (Vol^{′}) |
---|---|---|
0 | 0 | 3.3·10^{−6} |
0 | 1 | 1.2·10^{−6} |
1 | 0 | 9.5·10^{−5} |
1 | 1 | 2.5·10^{−5} |
Discussion
We developed the computational pipeline that can be used for model-to-model comparison in terms of the robustness to the perturbation of their kinetic parameters. Our work is based on the already established ’glocal’ method introduced by Hafner et al. [13]. In the first step, our approach roughly estimates global viable solution space through the optimization with the proposed genetic algorithm. Next, the viable solution space is more thoroughly explored with efficient local sampling. Because the viable solution space can be hyzz‘pothetically unconnected, we took a step further and proposed clustering to perform fine-grained exploration. The size and shape of viable regions can then be utilized for the assessment of model robustness. We successfully applied and validated our methodology on three distinct models that exhibit oscillatory and/or bistable behavior. Our approach utilizes exhaustive search of solution space, first by GAs and then with a prudent selection of samples, which is performed with local sampling in the direction of main principal components. We demonstrated the applicability of the proposed approach on three different deterministic ODE-based models.
One must also be aware of the potential drawbacks of the proposed approach. The first limitation is that the gap statistic and clustering are not perfect. There is not a strict consensus of what constitutes a separate cluster and different approaches will cluster the same population of samples differently. The drawback of gap statistic is that in order to test the null hypothesis, one must assume the distribution of the data points. We have shown, that if samples are uniformly distributed gap statistic always predicts the optimal number of clusters, however, this is not always the case for non-uniformly distributed data. To address this problem one could take methods for determining the number of clusters into account only partially as a guide and select the number of clusters based on observations. By our experience gap statistic consistently overestimated the number of clusters in the model of the AC-DC circuit and D flip-flop model. We approached this conservatively by setting the maximum number of clusters in the clustering step to 2 and by faster termination of the algorithm. The first viable region obtained by GAs can be clustered only once. All consecutive regions are then disregarded regardless of the gap statistic prediction.
The second possible drawback is that GAs and the probability based sampling introduces non-determinism into the sampling process, i.e. repetitive runs with the same configuration can return different results. We mitigated this with high population size and with a high probability of mutation and crossover in GA optimization. Similarly, we set the variance scaling factor λ relatively high at the beginning of our sampling process, when we were not as confident in the explored solution space, and gradually decreased it towards the end.
We have shown that the size and shape of a viable solution space are directly dependent on the definition of cost function and the threshold selection, which can be defined subjectively. We demonstrated the consequences of different interpretations of viable solution with the investigation of viable parameter regions on the same repressilator model for two different cost functions. The first cost function was defined in a strict manner, whereas the second one was defined more loosely. The viable parameter region of the repressilator model was greater when we employed the exploration of viable regions based on a loosely defined cost function. This coincides with our expectations and to some degree validates the correctness of our approach. In both cases, the parameter spans were roughly the same except for the degradation rate δ_{p} (see Fig. 6), which in the end contributed to the difference in the viable volume. Both cases have the same Hill coefficient ranges, 2≤n≤5. Higher values of Hill coefficient correspond to the positive cooperativity of transcription factors and a nonlinear response. The first implementation of a repressilator by Elowitz and Leibler [4] achieved non-linearity with the selection of transcription factors that function as oligomers, i.e. the TetR repressor protein is a dimer [33], and the LacI repressor is a tetramer [34].
To illustrate the exploration of viable regions with multiple well defined and unconnected clusters we analyzed the AC-DC model for two different modes of behavior. Our method proved to be effective in this scenario as well. Based on our observation, the AC-DC circuit tends more towards the oscillatory behavior than to bistability. We can assume this because the viable parameter volume is significantly larger for the parameter solutions that yield oscillatory dynamics in comparison to the bistable solutions. However, for the sake of simplicity, we excluded the signaling molecule S from the model and thus disregarded the switch like dynamics. It has already been shown that by adjusting the strength and/or the concentration of the signaling molecule the AC-DC circuit can exhibit both modes of behavior for the same sets of parameters [29]. Our simulations clearly indicate that in order to achieve either bistability or oscillations, the system should reflect non-linear transcription response. This can be described with larger values of the Hill coefficient [35]. Other researchers have already addressed this problem. For example, Lebar et al. [36] designed a bistable genetic switch based on designable DNA-binding domains of transcription-activator-like effectors (TALEs) [37]. Since TALEs bind non-cooperatively as monomers, a simple mutual repressor-based toggle switch does not support bistability. In order to introduce non-linearity and achieve bistability, Lebar et al. designed a bistable switch with an additional positive feedback loop of TALE repressors.
We investigated the robustness of the proposed D flip-flop model in dependence of different functional forms describing the protein degradation (Michaelian versus linear functions) and the transcription factor-promoter binding process (competitive versus independent binding). The flip-flop model proposed in [14] was extended into four different models upon which the proposed methodology was executed. We used the obtained results to assess the relative volumes of feasible solutions spaces in each of the models. These were used to perform the model-to-model comparison and to assess their robustness. We showed that Michaelian degradation form increases the chances of obtaining oscillations as already described in [38], but not as much as noncompetitive binding sites at promoter level [39]. When performing stochastic simulations we observed larger noise sensitivity as in the repressilator or AC-DC ciruit models. Solutions more resilient to intrinsic noise could potentially be obtained with the selection of different cost function defining feasibility of the solution, i.e. with larger oscillation amplitudes. The required amplitudes used in the D flip-flop model were set to 50 nM while amplitudes between 200 and 400 nM were regarded as feasible in the repressilator and AC-DC circuit models. Our results still indicate that the D flip-flop is robust to perturbations of its kinetic parameters and that the possibility of its implementation in the biological host is promising. Andrews et al. [30] demonstrated the feasibility of D flip-flop in E. coli. Still, their flip-flop is triggered on high signal levels and not on an edge of the synchronization signal. This makes the circuit hard to control, since the high level of the synchronization signal needs to be long enough to trigger the transition from one state to another and at the same time short enough to prevent multiple switches.
Our results indicate that the viable solution space of biological oscillators is generally well defined and connected, which has been already confirmed by other researchers [13, 17]. This is expected for the naturally occurring motifs that exhibit oscillations since they possibly evolved with random mutations that contributed to the small gradual changes of kinetic parameters [40]. However, it is interesting to observe similar properties in the synthetic circuits. This can be partially explained with the fact that the design is to some degree inspired by the systems we can observe in nature. Except for the AC-DC circuit, our models displayed single connected viable regions. The existence of multiple unconnected viable regions for the AC-DC circuit can be contributed to its capability of multimodal behavior. The values parameters span are generally different for both modes of behavior.
For a biological system to be robust, it must be able to withstand the fluctuations of biochemical parameters due to external factors, intrinsic noise, and single-cell variability. Our methodology can give one the insight into the shape and size of the viable parameter regions, and into the overall robustness of a system. Our approach has, therefore, two main applications. Firstly, by knowing the effect of parameters on system behavior, one could fine-tune the problematic parameters and use synthetic constructs, such as degradation tags to speed-up protein degradation [41], or design parts with higher binding affinities. For example, Fink et al. [7] designed coiled coils to increase the affinity between split proteases and thus increased the response of a system. However, approaches to experimentally tune the value of a given parameter are quite limited, especially in a predictive way. Secondly, one could use the proposed methodology to compare different systems with similar behavior and different topologies. For example, simple bistable switch could be compared with the bistable switch with positive feedback loops proposed by Lebar et al. [36]. The results of these comparisons could guide the researcher in the selection of more robust topologies and finally in the process of the implementation of reliable biological circuits.
We validated our methodology on biological GRN models that primarily exhibit oscillatory or bistable behavior. However, it is not hard to see how one could adapt this approach to cover other modes of behavior as well. By modifying the genetic algorithm and adjusting the cost function, one can adapt our approach to a variety of dynamical models and not only models of GRNs. We demonstrated the application of the proposed methodology solely on the results obtained with deterministic simulations. These models describe the average response of the system without the noise influences [42]. The noise influences can be to some degree indirectly analyzed with the analysis of parameter variability effects on the deterministic dynamics of the system [43]. In many cases it is more suitable to use stochastic modeling approaches, such as SSA [26]. These approaches directly describe the inherent stochasticity of biochemical reactions. In our case studies we only used the stochastic simulations to validate the correctness of the results obtained with deterministic approaches. However, these approaches could as well be used to generate the data upon which the proposed methodology would be applied. Moreover, the methodology could be used in a combination with any other either experimental or computational approach that is able to generate the response of the system at the sampled parameter values. However, there are some potential drawbacks in the straightforward application of these approaches. Deterministic models will always yield the same response for a given set of initial conditions and parameter values. Contrary, stochastic or experimental procedures will always differ to some degree even with the same conditions [27]. This means that multiple repetitions of the same experiment will be needed to achieve statistical significance, which in turn increases the time complexity of our methodology. Most of the approaches that present an alternative to deterministic models inherently increase the time complexity to generate the results even for a single simulation run. In order to at least partially circumvent this problem, one of the many parallelized adaptations of SSA with lower computational complexity could be applied [44].
Conclusion
In this paper we proposed a novel approach that can be used to assess the viable parameter regions for an arbitrary GRN model, and which can be applied in the design of synthetic biological systems. Identified parameter regions allow us to compare different models in terms of their robustness and also to identify the critical segments of the selected system. This can be exploited for the design of more reliable and robust systems. For example, if the degradation rates of observed proteins are constrained to a small interval, one can then specifically focus on this segment by exploring the effects of different degradation tags [41] and thus fine-tune the problematic parameters. Moreover, our approach can be used as a foundation for other analyzes. For example, bifurcation or sensitivity analysis can be done more efficiently and with higher precision if one is confident in the size and shape of the viable solution space.
Notes
Acknowledgements
Results presented here are in the scope of the Ph.D. thesis that is being prepared by Žiga Pušnik.
Authors’ contributions
ŽP designed the method, performed the experiments, and analyzed the results. ŽP and MMo devised the study and wrote the manuscript. MMo supervised the study. MMo, MMr and NZ provided critical feedback and helped shape the research, analysis and manuscript. All authors read and approved the final manuscript.
Funding
The research was partially supported by the scientific-research programme Pervasive Computing (P2-0359) financed by the Slovenian Research Agency in the years from 2017 to 2023, by the basic research project CholesteROR in metabolic liver diseases (J1-9176) financed by the Slovenian Research Agency in the years from 2018 to 2021, and by the infrastructure grant ELIXIR supported by MRIC UL (grant number I0-0022).
Ethics approval and consent to participate
Not applicable.
Consent for publication
Not applicable.
Competing interests
The authors declare that they have no competing interests.
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