Combined Model of Intrinsic and Extrinsic Variability for Computational Network Design with Application to Synthetic Biology
[['Biological systems are complex , inherently noisy and only partially understood [1]–[5] .', 'Systems and synthetic biologists are striving to better understand these systems , as well as to discover generally applicable principles for controlling them in biomedical and biotechnological applications .', 'For example , a branch of systems biology studies how to best interfere with variable and under-characterized signaling pathways to identify novel drug targets [6]–[8] .', 'In clinical pharmacology , decisions need to be made by using uncertain models of drug effects on the human body across populations and should ideally be robust to patient-to-patient differences [9] , [10] .', 'Synthetic biologists strive to build synthetic circuits that perform desired functions across a population of cells , despite their noisy nature , cell-to-cell variability , and changing environments [11] , [12] .', 'We are faced with a challenge of how to best represent , simulate , analyze , and carry out design for noisy systems with under-characterized biochemical properties , which are often represented as uncertain parameters .', 'We propose a modeling and simulation framework as a tool to aid in meeting these challenges .', 'In this manuscript we specifically focus on making design decisions for synthetic genetic networks , but the modeling technique is sufficiently general to be applicable to a wide range of problems in biology , biological engineering , and medicine .', 'The framework accounts for different sources of variability and is computationally efficient , so that it allows screening across broad parameter ranges .', 'Intrinsic variability ( or intrinsic stochasticity ) is a relatively well understood aspect of biological models .', 'It arises from the probabilistic nature of the timing of collision events between reacting biological molecules , and its effect is most pronounced when the number of molecules in the system is small .', 'Traditionally intrinsic variability is modeled by a stochastic master equation , which is the foundation for modeling stochastic dynamics in most physical , chemical , and biological phenomena [13] .', 'Unfortunately , its analytic solution can only be found for a few trivial models , and a good alternative for studying stochastic models is the exact simulation framework of Gillespie [14] .', 'However , to obtain a distribution resulting from the intrinsic variability , many trajectories of the Gillespie algorithm need to be simulated , which can be computationally expensive .', 'A practical alternative is to use analytically tractable approximation schemes [15] , [16] .', "In this manuscript we approximate stochastic dynamics by van Kampen's -expansion [13] ( also called the linear noise approximation or perturbation expansion; see Methods ) , for its computational efficiency and analytic form .", 'The -expansion separates the macroscopic dynamics from the fluctuations around it , describing each of these parts by a set of ordinary differential equations ( ODEs ) .', 'This model allows for efficient propagation of the first two moments of the intrinsic noise distribution through time using deterministic equations for mean , variances , and covariances .', 'The model very accurately approximates stochastic dynamics for medium and large molecular numbers and when variability is small compared to the mean number of molecules , but can lose on accuracy when numbers of molecules are very small and the relative fluctuation size increases [17]–[19]; here we check using Gillespie simulations that the -expansion distributions are accurate .', 'Analytic studies of the accuracy of the -expansion and comparison with other traditional approaches for modeling intrinsic noise such as Fokker Planck equation and the chemical Langevin equation can be found in [17] , [18] , [20] and a summary on validity of the -expansion in the Supplementary material of [19] .', 'Despite this caveat , and mainly due to its efficiency and possibility of analytic study , the -expansion has played an important role in advancing the understanding of intrinsic variability [3] , [18] , [21]–[24] .', 'The -expansion can most conservatively be applied to models with a single steady state — in this manuscript we will only consider such models — but with certain limitations or modifications it can also be applied to multimodal and oscillatory models [22] , [25]–[27] .', 'Significant effort has been invested in modeling intrinsic variability in systems and synthetic biology , although it has been shown that extrinsic variability generally dominates , especially in eukaryotic systems [2] , [28] .', 'Extrinsic variability arises due to varying components upstream of the system of interest; these components affect the system , varying stochastically in time themselves , and might be present in different amounts in cells due to differences such as size and stage of the cell cycle [2] , [3] , [29] .', 'For example , the numbers of ribosomes and the numbers of RNA polymerases vary in time and between cells .', 'Another source of extrinsic variability is cell-to-cell variability of the gene copy number , which is common in synthetic biology applications when genes are delivered into cells by plasmid transfection , after which different numbers of plasmids are taken up by different cells .', 'As a result of such sources of variability , single cells within a population possess distinct quantitative dynamic behaviors .', 'As yet , there is no commonly accepted framework for modeling extrinsic variability .', 'Despite several strong mathematical and theoretical studies of intrinsic and extrinsic variability [2] , [3] , [30] , [31] , computational modeling efforts that combine intrinsic and extrinsic variability are still rare .', 'Shahrezaei et al . [32] proposed an extension to the Gillespie approach that includes kinetic parameter perturbations representing extrinsic variability; the downside of this method is that it is extremely costly .', 'Scott et al . [26] proposed a more efficient , approximate model for steady-state extrinsic variability that can account for variations of one parameter at a time .', 'Zechner et al . [33] used low-order moments through the moment closure approach to approximate intrinsic and extrinsic distributions; this approach requires analytic derivation of a new model structure for each additional extrinsic factor .', 'Hallen et al . [34] proposed a non-mechanistic method of modeling extrinsic variability by perturbing the steady-state intrinsic noise distribution , but without any mechanistic assumption regarding the sources of extrinsic variability .', 'Singh et al . [35] model extrinsic variability by adding noisy exogenous signals to an intrinsic stochastic model .', 'Here we model extrinsic variability by introducing variability in model parameters and initial conditions; rather than considering them as point values , we consider them as distributions ( Figures 1 , S1 ) .', 'We propagate these distributions through a model to simulate model output distributions resulting from extrinsic variability .', 'For computational convenience we work with normally distributed parameters , ( for simplicity denotes a vector of all parameters and initial conditions ) .', 'To simulate propagation of extrinsic variability through the model , we use the Unscented Transform ( UT ) .', 'The UT efficiently maps the first two moments of the variability distribution in the parameter space onto the first two moments of variability distribution in the output .', 'Estimates of the mean and covariance matrix obtained by the UT are accurate to second order in the Taylor series expansion for any nonlinear function , which makes the algorithm very appealing for propagating distributions through nonlinear functions [36] .', 'Nonlinearity is propagated through simulating the nonlinear function for a chosen set of parameters ( called sigma-points , see Methods ) and reconstructing the output distribution from these individual simulations .', 'This can capture nonlinearity such as a shifts in a mean , for example , when extrinsic variability increases .', 'Despite knowing that extrinsic variability contributes significantly to the total variability , little is known about its sources [37] .', 'This creates a major caveat that arises when attempting to make informed design decisions .', 'The second ubiquitous caveat on the way to building predictive models of stochastic systems is that kinetic parameters are often unknown .', 'Here we are motivated by a question of how to make robust design choices , given these uncertainties and limited knowledge of extrinsic variability .', 'In this paper we use our method to study variability in simple gene regulatory networks; such networks are a basis for transcription-based synthetic circuits .', 'We first introduce a combined intrinsic and extrinsic modeling approach and derive the expression for the total variability .', 'The framework is then used to explore the effect of self-repression on protein variability in simple genetic networks that are simultaneously under different sources of noise .', 'We are interested in details related to how self-repression can be used as an element in synthetic circuits to reduce variability .', 'We consider two types of self-repression , on a transcriptional and on a post-transcriptional level , and ask which is more successful in reducing protein variability .', 'We are further interested in specific design principles that help optimally achieve the aim of noise suppression .'], ['The central methodological advance of this manuscript is a method and efficient framework to model total variability as a combination of intrinsic and extrinsic sources .', 'Here we formalize the framework ( overview in Figure 1 ) and illustrate it on a simple model of transcription and translation of a single gene , with species mRNA , , and protein , ( Figure 2A ) [38] .', 'The parameters of the model are transcription rate , , translation rate , , and mRNA and protein degradation rates , and , respectively .', 'is the initial condition representing the copy number of genes encoding protein .', 'We denote the vector of parameters and initial conditions by .', 'Here we represent intrinsic variability through the -expansion , which provides a set of ordinary differential equations for the concentration means that are identical to a traditional ODE model without variability , plus an additional set of differential equations that describe time derivatives of the individual variances and covariances ( Figure 2B ) .', 'All equations together approximate the time evolution of the intrinsic noise distribution .', 'Note that the ODEs for the means do not depend on the variances and covariances but that the ODEs for the variances and covariances depend on means , variances , and covariances .', 'Note also that ODEs for propagating the variance and covariance do not introduce any additional rate constant parameters beyond those needed to propagate the concentration means , but that initial values for variance and covariance need to be introduced in order to integrate their differential equations .', 'We express this as a system of ordinary differential equations ( 1 ) ( 2 ) where is a vector representing the mean numbers of mRNA and protein and is the symmetric covariance matrix We simulated intrinsic variability in single gene expression using the -expansion model .', 'Time-course trajectories of mean number of mRNA and protein ( red ) and their variances ( green ) are shown in Figure 2C .', 'The Figure also shows individual trajectories from stochastic simulation runs ( blue ) using the Gillespie algorithm ( Methods ) [14] , which are consistent with the -expansion simulations .', 'A more detailed comparison shows that the distributions from the -expansion and the stochastic Gillespie simulations are nearly identical ( Figure S2 ) .', "The multivariate Gaussian distribution is depicted by an ellipse with mean whose axes' directions are determined by eigenvectors and their sizes by eigenvalues of the covariance matrix ( Figure 2D , see Methods for further details ) .", 'This represents variability in mRNA and protein counts due to intrinsic noise only .', 'We next introduced extrinsic variability into the model , by introducing parameters that follow a distribution with a probability density function , rather than parameters fixed to a particular value .', 'Operationally , the combined distribution is a superposition of intrinsic distributions for different parameter realizations sampled from the underlying parameter probability distribution , ( Figure 2E ) .', 'The resulting output distribution represents the total variability ( i . e . , variability resulting from intrinsic as well as extrinsic sources ) .', 'Mathematically we represent this distribution with a mixture model with probability density function ( 3 ) where is a probability density of the intrinsic noise distribution , in our specific case a normal distribution with mean and a covariance matrix resulting from the -expansion model with parameters .', 'For single gene expression , the mixture model readsand its steady state is schematically depicted in Figure 2E .', 'This framework allows us to calculate the mean and variance of the total variability distribution; the mean of a random variable drawn from the mixture model ( 3 ) was calculated as ( 4 ) and by using the law of total variance [39] we obtained the covariance matrix [30] ( 5 ) This equation shows that the total variability is decomposable into two parts: variability due to intrinsic stochasticity , , and variability due to extrinsic sources , ( Figure 2F ) .', 'So far we have not restricted ourselves to any specific form of the parameter distribution ; in order to obtain a combined model and calculate the total variability , we generally need to exhaustively draw samples from and simulate the -expansion model for each sampled parameter .', 'However , for normally distributed parameters , , we can take a considerably more efficient approach , the Unscented Transform ( UT , see Methods ) .', 'The UT selects a small collection of representative control points in the parameter space ( called sigma points ) , propagates them through the -expansion simulation model to obtain the output instance for each , and then reconstructs the first two moments of the output distribution ( which is equivalent to the above mixture model distribution ) by appropriate weighting of the sigma points .', 'From the output distribution , the total mean and the total variance are easily computed .', 'See Text S1 for further details and an example for the single gene expression model .', 'We checked that our novel combined simulation framework accurately approximates the exact total protein and mRNA distributions resulting from Gillespie simulation .', 'We first introduced extrinsic variability into one parameter only , and then into four parameters .', 'In the first case , approximate distributions resulting from the novel framework fit very well to exact distributions in both steady and transient states ( Figure S3 ) .', 'In the second example with extrinsic variability in four parameters , the approximation was slightly worse; this is becasue the total variability distribution was not Gaussian , but skewed , and perhaps also to inexactness in the -expansion ( Figure S4 ) .', 'However , our approximation approximated well the distribution up to the second moment , i . e . the mean and the variance .', 'Figure 3 shows examples of interplay between intrinsic and extrinsic variability for distinct amounts of extrinsic variability .', 'In this example we considered parameters to be independent ( i . e . , the covariance matrix had zero for all off-diagonal entries ) , and all parameters were varied with the same coefficient of variation ( see Methods ) .', 'When the variability in parameters was low , intrinsic variability was dominant in the system ( Figure 3A ) ; this can be seen from the sizes of the green ellipses ( the “average” of the green ellipses corresponds to in Eq . 5 ) , which are much larger than the size of the red ellipse ( corresponding to ) .', 'On the other hand , for high parameter variability the extrinsic variability became dominant ( Figure 3F; red ellipse is larger than the green ones ) .', 'For intermediate parameter variability , both intrinsic and extrinsic sources contributed comparably to the overall variability .', 'Notice also that with increasing extrinsic variability , the intrinsic components changed their size and shape; this corresponds to the observation that intrinsic variability can change with kinetic parameters [40] .', 'The total variability is the combination of red and averaged green ellipses; with increasing extrinsic variability , the variability of the output increased .', 'High extrinsic variability has also shifted the mean; for example protein mean has increased when extrinsic variability was increased ( this can be seen by comparing the x-component of the red ellipse mean on Figures 3A and 3F , where the mean of the red ellipse represents the mean of the output – combined intrinsic and extrinsic noise – distribution ) .', 'In the remainder of this paper we report results obtained by using this framework to study how different topologies of self-repressing genetic networks affect the total variability in proteins .', 'Negative feedback and self-repression are recurrent motifs observed in gene regulatory networks .', 'Their potential role in suppressing variability has identified them as promising design elements in synthetic biology [32] , [41]–[48] .', 'Here we studied the effects of self-repression on variability of the output protein in a single gene model , while taking into account both intrinsic and extrinsic sources of variability .', 'We compared a case of negative autoregulation on a transcriptional level with a case of post-transcriptional regulation .', 'In the transcriptional autoregulation case , the output protein acts as a repressor that multimerizes and binds to its own promoter region to repress its own transcription ( Figure 4A ) .', 'In the post-transcriptional case , we model synthetic microRNA ( miRNA ) that binds to its cognate mRNA , simultaneously preventing translation and promoting degradation ( Figure 5 ) .', 'In the remainder of this section we explore the effectiveness of these control mechanisms .', 'The results provide specific guidelines for designing transcriptional and post-transcriptional autoregulatory networks for achieving best ( or worst ) suppression of protein variability .'], ['Variability is ubiquitous to biological systems , and yet it is not well understood .', 'In order to engineer and intervene in such systems successfully , variability must be taken into account in planning and interpreting experiments and in designing synthetic networks .', 'Here we have introduced a framework for modeling and simulation of dynamical systems under both intrinsic and extrinsic sources of variability .', 'One reason that understanding and treating variability is so especially important for synthetic biology is that one seeks to design circuits and devices that can perform successfully across a population of cells and variety of experimental conditions; without understanding variability and fluctuations , one risks building devices or designing therapies that only perform well in a fraction of cells ( or patients ) or only under limited environmental conditions .', 'Here we addressed the problem of constructing circuits that exhibit low variability across cells .', 'We investigated transcriptional and post-transcriptional designs of self repression , and studied their relative promise for noise suppression .', 'We explored their behavior for a range of feedback strengths and other design parameters , and for various hypothetical amounts and types of extrinsic variability .', 'We derived a set of design principles for best noise suppression in a protein concentration of interest .', 'We showed that transcriptional autoregulation is more successful than post-transcriptional in suppressing noise under a wide range of intrinsic and extrinsic variability levels and conditions .', 'The following design principles were shown to best suppress protein concentration variability:', '( i ) high cooperativity in protein binding to DNA and low cooperativity in miRNA binding to mRNA ,', '( ii ) imperfect complementarity between miRNA and mRNA was preferred to perfect complementarity , and', '( iii ) correlated expression of mRNA and miRNA — for example , on the same transcript .', 'We further showed that correlations in kinetic parameters between cells affected the ability to suppress variability , and that variability in transient states did not necessarily follow the same principles as variability in the steady state .', 'Also , we note that biochemical design can not avoid fundamental limits on the lower bound of stochastic fluctuations [57] .', 'In this work we have focused on one objective , which is to decrease variability of protein expression .', 'In synthetic biology applications , one might be interested in additional objectives , such as maximizing the mean protein expression , or keeping the mean fixed while suppressing variability etc .', 'The same analysis can be performed for such objectives , and could potentially lead to different guidelines .', 'Our novel modeling and simulation framework combines intrinsic and extrinsic sources of variability by combining the -expansion with the unscented transform .', 'In contrast to few previous methods , our framework is based on a deterministic simulation with ordinary differential equations .', 'One advantage of our method is that it incorporates extrinsic variability by repeated simulation of the same intrinsic model for carefully selected parameter combinations where the number of simulations increases linearly with the number of extrinsic factors ( see Methods ) , rather than generating a new model and exponentially increasing the model size for each new extrinsic factor [33] .', 'This not only increases computational efficiency , but makes possible analysis with a wide variety of available tools .', 'It also allows for screening across a large range of parameters ( such as feedback strength , and rates , and unknown sources and quantities of extrinsic variability ) .', 'The new approach is well suited to support the design of synthetic circuits from building blocks with partially unknown kinetic properties but that will perform desired functions robustly in variable intracellular and extracellular environments and across a population of cells .', 'An important aspect of the framework is that it can be applied to study variability and dynamics at transient as well as steady states; we have shown that variability might behave differently in both regimes , which reiterates the importance of developing and using tools that are applicable beyond steady-state regimes .', 'The computational efficiency of the framework , however , comes at a cost of approximation; macroscopic dynamics remain non-linear , but intrinsic and extrinsic variability distributions are both approximated by the first two moments .', 'We have shown that transient and steady-state distributions resulting from our approximation framework closely fit to exact distributions simulated by a Gillespie algorithm and parameter sampling .', 'We note once more that the -expansion is derived under the assumption of large numbers of molecules and we recommend the results to be tested by more accurate methods , especially for low molecule numbers .', 'For example , a sufficient number of Gillespie runs provides the exact distribution [14]; another powerful technique , the Langevin approximation , can be used to obtain more accurate than -expansion [20] , although still approximate estimates of the variance at a steady state , and its disadvantage is that it requires monitoring of the convergence .', 'An important point , however , is that we have focused less on calculating exact quantitative values of variability , and more on what changes in the network drive variability increases or decreases .', 'Our approximation framework is sufficient for addressing this question , and due to computational efficiency in both steady and transient states it is even advantageous compared to exact simulation .', 'We have applied the framework to systems with a single steady state , but there is scope for extending the framework to oscillatory and multimodal systems [25]–[27] .', 'Derivation of improved versions of the -expansion is an area of active research , for example accounting for slow and fast variables [58] or increasing accuracy for small number of molecules [59] .', 'By developing better approximation frameworks for intrinsic noise modeling , our combined intrinsic-extrinsic method will increase in accuracy , too .', 'Furthermore , while the dynamics originating from intrinsic variability was modelled by the -expansion approximation framework in this manuscript , a similar framework could be constructed based on other approximation modeling approaches for intrinsic variability , for example the moment closure method [60] or mass fluctuation kinetics [16] .', 'The extrinsic variability modeled through parameter distributions in this manuscript was assumed to be constant in time ( i . e . , static ) , and a natural extension would be to introduce time-varying parameter distributions ( i . e . , dynamic ) [31] , [61] .', 'Because of this , for example , our method in its current form can not be used to quantify the dependence of the total output variation on the lifetime of extrinsic fluctuations [32] .', 'There are different sources of biological variability and uncertainty; a careful interpretation and appropriate inclusion in a mathematical model is crucial to elucidate the dynamics of biological variability and understanding of these different levels of biological complexity .', 'Firstly , there are intrinsic and extrinsic sources of variability ( discussed at length above ) .', 'Secondly , there is parameter uncertainty , which encompasses our limited knowledge of kinetic rate values .', 'This uncertainty is commonly included in biological models and simulations as a distribution of kinetic values , however , it is important to treat it separately from extrinsic variability , which is also modeled by treating parameters as distributions .', 'An example of appropriate mathematical treatment is a hierarchical Bayesian model [33] , [62] .', 'A further type of uncertainty is model uncertainty , by which we refer to our uncertainty in model topology; the present manuscript does not address this type of uncertainty .', 'And thirdly , when modeling or analyzing experimental biological data , one also needs to account for measurement noise ( or measurement error ) .', 'In order to infer sources of variability and reduce uncertainty in parameters , there is a need for inference techniques for models that account for these different levels of biological variability , uncertainty , and measurement noise .', 'The presented modeling framework is only the first step in the systems biology cycle of modeling , designing experiments , and updating the model by inference from collected data .', 'Based on this modeling framework , we are now developing tools that account for variability also in the experimental design stage , and for updating the models from noisy data through parameter inference and model selection algorithms .', 'These techniques will allow us to better explore and account for biological variability in synthetic biology , a ubiquitous aspect of biological systems that is in practice often neglected .'], ['The -expansion is an approximation of the master equation .', 'It separates the macroscopic part of dynamics from the fluctuations around it , describing each of these parts by a set of ODEs [13] .', 'The resulting -expansion model approximates the first two moments of the intrinsic noise distribution in the form of ODEs for each mean , variance , and covariance; for species in the model , the -expansion generates equations for the means , equations for variances , and equations for covariances between all pairs of species .', 'Readers interested in a theoretical derivation of the -expansion modeling framework are referred to van Kampen [13] .', 'As a brief summary we note that the -expansion is derived assuming that the macroscopic part of dynamics can be separated from the fluctuations centred around it , and that the fluctuations scale as the square root of the number of molecules [63] .', 'This is captured by the following ansatz: , where is the number of molecules , is the macroscopic concentration and the fluctuations .', 'Parameter represents the system size and can be thought of as proportional to the volume .', 'This ansatz is introduced into the master equation governing the evolution of probability distribution and then the master equation is Taylor-expanded around .', 'The terms of order collected together determine equations governing macroscopic dynamics , and the terms of order the equations for dynamics of fluctuations .', 'The resulting set of ordinary differential equations describing macroscopic ( or average ) behavior is ( 8 ) where is the stoichiometry matrix and are the reaction propensities of reactions .', 'The ODE for the covariance matrix of the fluctuations , , centred around is ( 9 ) where is a Jacobian of a deterministic system given by equation ( 8 ) , evaluated along the macroscopic trajectory , and is called the drift .', 'Matrix is the diffusion matrix , .', 'Equations ( 8 ) and ( 9 ) together form the -expansion model , which approximates the first two moments ( mean and covariance matrix ) of the distribution that solves the master equation .', 'The -expansion model is a deterministic and approximate description of intrinsic noise distribution dynamics in time .', 'We visualised the multivariate normal distributions as ellipses .', 'The ellipse is one of the contours that represent equal probability mass of the distribution .', 'In general , these ellipses are constructed from eigenvalues and eigenvectors of the covariance matrix .', 'They are centred at mean , their axis directions are determined by the eigenvectors and axis sizes by the eigenvalues , .', 'Different values of specify different contours .', 'In our figures we choose unless otherwise specified .', 'In particular , the ellipse determined by equationrepresents the multivariate analogue of the “confidence interval” containing of the probability mass of the distribution .', 'For example , a bivariate ( ) normal distribution represented by an ellipse with axes lengths , , contains of distribution , and an ellipse containing of a distribution has axes of lengths , .', 'The unscented transform ( UT ) is an algorithm that efficiently propagates normally distributed inputs through a nonlinear function to obtain a distribution on the function outputs .', 'Figure 13 provides an overview of the UT .', 'Here was the multivariate normal distribution on the space of parameters and initial conditions of dimension .', 'We deterministically sampled from a set of sigma points ( 10a ) ( 10b ) ( 10c ) and assigned weights ( 11a ) ( 11b ) ( 11c ) where .', 'See ref .', '[36] for guidelines on how to choose parameters , and ; we chose , , .', 'The above weights define the scaled unscented transform , which is the generalized version of the unscented transform algorithm [64] .', 'Function was evaluated for every sigma point , The model outputs obtained were then reconstructed into a multivariate normal distribution with mean and covariance matrix determined by the following equations: ( 12a ) ( 12b ) ( 12c ) We note that the UT is computationally much more for efficient compared to random sampling; UT only requires 2L+1 simulations ( one for each sigma point ) , while this number needs to be considerably higher for exhaustive sampling , especially for high dimensional parameter spaces .', 'In terms of accuracy , the UT estimates the first two moments accurately to second order in the Taylor series expansion for any nonlinear function [65] , [66] .', 'However , should one sample 2L+1 points randomly and propagate them , one will not achieve this accuracy and therefore exhaustive sampling is needed to achieve the same level of accuracy as the UT ( the exact number of simulations needed depends on the specific system and space dimension of parameters and initial condit
['Introduction', 'Results', 'Discussion', 'Methods']
['Biological systems are inherently variable , with their dynamics influenced by intrinsic and extrinsic sources .', 'These systems are often only partially characterized , with large uncertainties about specific sources of extrinsic variability and biochemical properties .', 'Moreover , it is not yet well understood how different sources of variability combine and affect biological systems in concert .', 'To successfully design biomedical therapies or synthetic circuits with robust performance , it is crucial to account for uncertainty and effects of variability .', 'Here we introduce an efficient modeling and simulation framework to study systems that are simultaneously subject to multiple sources of variability , and apply it to make design decisions on small genetic networks that play a role of basic design elements of synthetic circuits .', 'Specifically , the framework was used to explore the effect of transcriptional and post-transcriptional autoregulation on fluctuations in protein expression in simple genetic networks .', 'We found that autoregulation could either suppress or increase the output variability , depending on specific noise sources and network parameters .', 'We showed that transcriptional autoregulation was more successful than post-transcriptional in suppressing variability across a wide range of intrinsic and extrinsic magnitudes and sources .', 'We derived the following design principles to guide the design of circuits that best suppress variability:', '( i ) high protein cooperativity and low miRNA cooperativity ,', '( ii ) imperfect complementarity between miRNA and mRNA was preferred to perfect complementarity , and', '( iii ) correlated expression of mRNA and miRNA – for example , on the same transcript – was best for suppression of protein variability .', 'Results further showed that correlations in kinetic parameters between cells affected the ability to suppress variability , and that variability in transient states did not necessarily follow the same principles as variability in the steady state .', 'Our model and findings provide a general framework to guide design principles in synthetic biology .']
['Variability is inherent in biological systems , and in order to understand them , we need to be able to model different sources of variability .', 'Systems have evolved to harness and control the variability , and more recently , synthetic biologists are trying to learn how to control variability in engineered biological systems .', 'Several sources of variability exist; they arise due to stochastic expression of genes , which is most pronounced when numbers of mRNA and protein molecules are low , as well as due to differences between individual cells .', 'Here we propose a modeling framework that combines different sources of biological variability .', 'Furthermore , current research seeks to control biological variability though robust design of synthetic biological circuits , for example for use in therapies and other biomedical or biotechnological applications .', 'Here we apply our framework to guide design of synthetic circuits that use transcriptional and post-transcriptional regulation to suppress variability in the output protein of interest .', 'We find that certain properties and network designs are better than others in their ability to control variability , and here we report on the design guidelines to aid synthetic circuit design to suppress variability , in spite of our uncertain knowledge of parameters .']
['biotechnology', 'bioengineering', 'systems biology', 'biochemical simulations', 'biological systems engineering', 'synthetic biology', 'biology', 'computational biology', 'molecular biology', 'genetics and genomics', 'engineering']