Estimating confidence intervals in predicted responses for oscillatory biological models
© St. John and Doyle; licensee BioMed Central Ltd. 2013
Received: 5 January 2013
Accepted: 15 July 2013
Published: 29 July 2013
The dynamics of gene regulation play a crucial role in a cellular control: allowing the cell to express the right proteins to meet changing needs. Some needs, such as correctly anticipating the day-night cycle, require complicated oscillatory features. In the analysis of gene regulatory networks, mathematical models are frequently used to understand how a network’s structure enables it to respond appropriately to external inputs. These models typically consist of a set of ordinary differential equations, describing a network of biochemical reactions, and unknown kinetic parameters, chosen such that the model best captures experimental data. However, since a model’s parameter values are uncertain, and since dynamic responses to inputs are highly parameter-dependent, it is difficult to assess the confidence associated with these in silico predictions. In particular, models with complex dynamics - such as oscillations - must be fit with computationally expensive global optimization routines, and cannot take advantage of existing measures of identifiability. Despite their difficulty to model mathematically, limit cycle oscillations play a key role in many biological processes, including cell cycling, metabolism, neuron firing, and circadian rhythms.
In this study, we employ an efficient parameter estimation technique to enable a bootstrap uncertainty analysis for limit cycle models. Since the primary role of systems biology models is the insight they provide on responses to rate perturbations, we extend our uncertainty analysis to include first order sensitivity coefficients. Using a literature model of circadian rhythms, we show how predictive precision is degraded with decreasing sample points and increasing relative error. Additionally, we show how this method can be used for model discrimination by comparing the output identifiability of two candidate model structures to published literature data.
Our method permits modellers of oscillatory systems to confidently show that a model’s dynamic characteristics follow directly from experimental data and model structure, relaxing assumptions on the particular parameters chosen. Ultimately, this work highlights the importance of continued collection of high-resolution data on gene and protein activity levels, as they allow the development of predictive mathematical models.
A cell’s behavior is governed by the dynamic and selective expression of its genes, in which each protein’s activity depends on a careful balance between transcription, translation, transport, and degradation rates. These rates, which change with environmental conditions and are often impossible to measure accurately in vivo or in vitro, determine the function of a regulatory pathway. While studying the roles of individual proteins can often provide some insight on how a particular function is achieved, this approach is limited in explaining complicated cellular phenomena at the scale of dozens to hundreds of interacting genes. With the aid of mathematical models, it is increasingly possible to create in silico realizations of genetic regulatory networks to examine their dynamic properties.
Essential to understanding how genetic circuits operate is connecting how inputs (i.e., environmental changes, extracellular signals) are processed to give the appropriate outputs (protein expression, cellular response). In some cases these quantities may be changes to oscillatory profiles: for example, seasonal changes in day length leading to flowering or hibernation. Models of genetic regulatory networks, often sets of ordinary differential equations (ODEs), contain many unknown parameters that must be estimated from experimental data. Derivatives of the model output with respect to changes in input, known as local sensitivities, are frequently validated experimentally or used to predict potential targets for pharmaceuticals. Since sensitivities can change drastically with respect to the particular parameter values chosen, the confidence associated with parameter and sensitivity values is an important consideration in model analysis and design.
Practical identifiability analysis is concerned with calculating confidence intervals in parameter estimates resulting from uncertainty in experimental data. Several techniques for such an analysis currently exist, and are commonly used in analyzing biological models[4–6]. In one method, the inverse of the Fisher information matrix is used to provide estimates of the variance in each parameter. However, since this method assumes a linearized model, the resulting symmetric normal distributions for each parameter do not accurately reflect the mapping of nonlinear models. In the bootstrap method, distributions in parameter estimates are found through optimum fits to repeated physical or in silico measurements. While accurate in finding the true nonlinear confidence intervals, this approach requires efficient and robust parameter estimation convergence.
Many systems biology models focus on describing interesting dynamic features from interlocked regulatory mechanisms. Limit cycle oscillations are common features in many biological networks, ranging from cell cycle control to cyclic firing of cardiac cells and circadian rhythms. In periodic systems, the behavior (and existence) of limit cycle oscillations is a discontinuous function of the parameters, complicating parameter estimation. Optimal values are traditionally found through trial-and-error type approaches[9, 10] or genetic algorithm search strategies, both of which are not amenable to bootstrap methods. Additionally, since the solutions are oscillatory, additional care must be taken in the calculation of the first-order sensitivity values. Here we calculate the sensitivity of the oscillatory period to parameter perturbation, a biologically relevant quantity that is often measured experimentally. Due to these complications, rigorous identifiability analyses of these models are typically not performed.
In this study, a bootstrap uncertainty analysis appropriate for oscillatory biological models is developed and applied to a previously published model of circadian rhythms. Circadian rhythms are near 24-hour endogenous oscillations in physiological processes found in many organisms, coordinated through transcription-translation networks with inherent time-delayed negative feedback[14–16]. In mammals, expression of circadian E box genes Period (Per) and Cryptochrome (Cry1 and Cry2) leads to elevated levels of their protein products, PER and CRY. The formation of a heterodimeric complex allows PER and CRY proteins enter the nucleus and subsequently suppress E-box mediated transcription, resulting in rhythmic gene expression. These networks serve as an excellent example of a functional genetic circuit, able to process subtle environmental cues while remaining robust to temperature variations and evolutionary disturbances. Accurate limit cycle models must capture not only the correct time-dependent dynamics, but also the correct input-output response. For circadian rhythms, high-throughput microarrays have provided high-resolution time-series data of gene expression levels. Additionally, knockdown experiments using RNA interference technology (siRNA) and small molecule modulators have resulted in a wealth of data on the dynamic responses to changes in key rates[13, 18–20]. This data, together with qualitative knowledge of the underlying network structure, permits the use and verification of a suitable uncertainty analysis.
To enable a bootstrap approach, we employ an efficient parameter estimation routine optimized for limit cycle models. Motivated by the increasing availability of high-resolution time-series measurements, we use an approach similar to multiple shooting, in which a nonlinear and discontinuous parameter estimation problem is transformed into a high-dimensional yet local optimization and solved via nonlinear programming. Since the desired shape of the limit cycle solution is known a priori, a relatively accurate initial guess for the parameters and trajectories can be found. By using multiple sets of in silico data of varying quality, we illustrate how error in experimental results is propagated to uncertainty in parameter sensitivity. Lower quality data - with either higher error or fewer sampling points - result in wider distributions of limit cycles and less identifiable responses. These results can be used in a priori experimental design, finding the minimum sampling points needed for an estimated experimental error to enable accurate modeling. Additionally, we show using literature data how this method can be used to discriminate between candidate model structures, revealing which one yields the highest predictive confidence.
Results and discussion
Here the period of oscillation is the smallest T>0 in which the equality (2) holds. Limit cycle oscillations are independent of the system’s initial values x(0), and are instead determined completely by the parameters p.
Here σ i j is the standard deviation associated with the measured mean of state i at time j. Using the data points to generate a suitable initial guess, parameter estimation may proceed via a nonlinear programming approach (see Methods, Additional file1). In this work, we assume that all states are measured to demonstrate how initial guesses can be generated directly from the input data. However, for systems with unmeasured states, initial guesses for the trajectory and parameter values can be provided by another approach, such as a global optimization routine. A bootstrap method was implemented by repeatedly sampling input data distributions to calculate a population of optimal parameter fits.
were calculated due to their independence of parameter magnitude[12, 23, 24]. Relative period sensitivities were integrated into the bootstrap method by calculating appropriate sensitivities for each estimated parameter set.
Effect of data quality on predictive confidence
Sensitivities that are experimentally distinguishable from zero are the most important for validation. Calculating a typical experimental value for a relative period sensitivity helps to calibrate which sensitivities might be verified experimentally. Referring to a recent RNA interference screen, periods changes of approximately 1 hour (5%) can be reliably measured using luminescence recordings. Assuming an increase in the corresponding mRNA degradation parameter value of 50%, this translates to a relative period sensitivity of 0.1. Thus, many of the identifiable values shown in Figures3–4 fall within the experimentally measurable range.
Application to literature data for model discrimination
The literature data used consisted of 7-8 concentration time points across a 24 hour period. Confidence intervals in the data were not available, so an optimistic 3% relative and 0.5% absolute error was assumed for each data point (). Figure5A shows the resulting time-series profiles for bootstrap estimations of each model. While additional kinetic parameters are typically thought to lower the predictive confidence of a model (the ‘curse of dimensionality’), the expanded model is able to better capture the oscillatory profiles with lower variability between solutions. Parameter and sensitivity distributions, Figure5B, similarly show how the expanded model parameterization is able to generate more confident predictions in model response. Since the resulting sensitivity identifiability for both models was relatively poor, we highlight sensitivities which pass a 90% confidence level threshold. These results thus indicate higher-resolution data on circadian components would help in conferring confidence to model predictions.
Two sensitivities, the PER translation rate (Figure5B,1) and the PER-CRY association rate (2), had high confidence and consistent direction in both the base and expanded parameterization - suggesting that the predicted values are robust to slight changes in both parameter value and model structure. Since a biological system can be modeled using many different combinations of kinetic assumptions, such a technique will likely prove useful in finding consistent predictions which are robust to slight differences in model equations.
Increasingly, mathematical models are being used to study biological systems where traditional experiments would prove infeasible. For example, in the search for drug targets, thousands of possible combinatorial perturbations can be quickly scanned for therapeutic effects using in silico modeling. This is especially useful in oscillatory systems with long periods, such as circadian rhythms, where a perturbed in vitro or in vivo system must be measured for multiple days before changes can be reliably determined.
However, since errors in model responses can arise from either incorrect structure or measurement noise, our confidence in in silico predictions is limited. Here we have developed a bootstrap approach suitable for periodic systems, and extended it to include uncertainty in predicted responses. With this method, errors due to local parameter effects can be identified, even in models with complicated dynamics. Furthermore, by considering multiple variations in model assumptions, we have demonstrated that a clearer result of trustworthy model predictions can be found.
Since this method takes advantage of time-series data to generate a strong initial guess for an otherwise difficult parameter estimation, it requires high-resolution data on the concentrations of all species in the model. In many biological systems, such data is only available for the activity levels of certain well-studied species. However, the continued development of high-throughput genomic and proteomic techniques promise to deliver time-series data for a much larger network of components. With expanding datasets, these methods will likely prove useful for the quantitative evaluation of uncertainty in larger biological models.
Generation of data for bootstrap methods
For each run, two thousand simulated measurements,, were generated from the true data,, using a normal distribution with and, in which ξ is the relative and η is the absolute error. Each simulated data set was then used to find a unique optimum parameter set, p⋆. Data sets that failed to converge, or reached a steady state solution (in which periodic sensitivities are undefined), were discarded from further analysis.
Since standard deviations in the data distributions were also used as optimization weights, a small amount of absolute error (η=0.001) was added to ensure errors in small values did not dominate the cost function.
Collocation methods and sensitivity analysis
In this work, the estimation of the unknown kinetic parameters is accomplished via nonlinear programming (NLP). In this method, we divide the limit cycle trajectory into finite elements, and approximate each with a degree Lagrange interpolating polynomial, x(t). The minimization of (3) proceeds by changing the state variable and parameter values, ensuring both periodic continuity and system dynamics. The number of variables of the NLP problem is therefore, where NEQ is the number of state variables and NP is the number of parameters. In the model considered in this study, with,, NEQ=8, and NP=21, the number of variables is 981. However, because suitable initial guesses can be found for both the state variable and parameter variables (see Figure1), relatively efficient convergence can be achieved. Detailed information on the algorithms used are presented in Additional file1.
Each parameter estimation took approximately 4 seconds on a 2.53GHz processor, with the subsequent limit cycle solution integration and sensitivity calculation taking approximately 0.5 seconds. Due to the parallel nature of the 2000 trials, computation times were alleviated by distributing the tasks onto a cluster of 160 compute nodes.
The numerical implementation of the nonlinear programming optimizations was accomplished using IPOPT. The CasADi computer algebra package was used to provide an interface to the IPOPT numerical libraries and supply derivatives to the cost and equality function calls through automatic differentiation.
Other libraries used were the SUNDIALS packages CVODES for ODE integration and KINSOL for the Newton iterations involved in the solution of the limit cycle. Integration of the sensitivity equations was performed by using the staggered-direct method from the CVODES integrator.
We thank Maria Rodriguez-Fernandez, Panagiota Foteinou, and Lukas Widmer for helpful discussions and critical readings of the manuscript; John Bushnell for technical support; and the Institute for Collaborative Biotechnologies for computing resources. This work was supported through the National Institutes of Health under award number 1R01GM096873-01, and by the Institute for Collaborative Biotechnologies through grant W911NF-09-0001 from the U.S. Army Research Office. The content of the information does not necessarily reflect the position or the policy of the Government, and no official endorsement should be inferred.
- Gutenkunst RN, Waterfall JJ, Casey FP, Brown KS, Myers CR, Sethna JP: Universally sloppy parameter sensitivities in systems biology models. PLoS Comput Biol. 2007, 3 (10): 1871-1878.PubMedView ArticleGoogle Scholar
- Knowles JD, Kell DB: The role of modeling in systems biology. System Modeling in, Cellular Biology: From Concepts to Nuts and Bolts. Edited by: Szallasi Z, Stelling J, Periwal V. 2006, Cambridge: MIT Press, 3-18.Google Scholar
- Raue A, Kreutz C, Maiwald T, Bachmann J, Schilling M, Klingmüller U, Timmer J: Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics. 2009, 25 (15): 1923-1929. 10.1093/bioinformatics/btp358.PubMedView ArticleGoogle Scholar
- Nihtilä M, Virkkunen J: Practical identifiability of growth and substrate consumption models. Biotechnol Bioeng. 1977, 19 (12): 1831-1850. 10.1002/bit.260191208.PubMedView ArticleGoogle Scholar
- Jiménez-Hornero JE, Santos-Dueñas IM, García-García I: Optimization of biotechnological processes. The acetic acid fermentation. Part II: Practical identifiability analysis and parameter estimation. Biochem Eng J. 2009, 45: 7-21. 10.1016/j.bej.2009.01.010.View ArticleGoogle Scholar
- Holmberg A: On the practical identifiability of microbial growth models incorporating Michaelis-Menten type nonlinearities. Math Biosci. 1982, 62: 23-43. 10.1016/0025-5564(82)90061-X.View ArticleGoogle Scholar
- Joshi M, Seidel-Morgenstern A, Kremling A: Exploiting the bootstrap method for quantifying parameter confidence intervals in dynamical systems. Metab Eng. 2006, 8 (5): 447-455. 10.1016/j.ymben.2006.04.003.PubMedView ArticleGoogle Scholar
- Goldbeter A, Keizer J: Biochemical Oscillations and Cellular Rhythms: The Molecular Bases of Periodic and Chaotic Behaviour Volume 51. 1996, Cambridge: Cambridge University PressView ArticleGoogle Scholar
- Forger DB, Peskin CS: A detailed predictive model of the mammalian circadian clock. Proc Natl Acad Sci USA. 2003, 100 (25): 14806-14811. 10.1073/pnas.2036281100.PubMedPubMed CentralView ArticleGoogle Scholar
- Leloup JC, Goldbeter A: Toward a detailed computational model for the mammalian circadian clock. Proc Natl Acad Sci USA. 2003, 100 (12): 7051-7056. 10.1073/pnas.1132112100.PubMedPubMed CentralView ArticleGoogle Scholar
- Mirsky HP, Liu AC, Welsh DK, Kay SA, Doyle III FJ: A model of the cell-autonomous mammalian circadian clock. Proc Natl Acad Sci USA. 2009, 106 (27): 11107-11112. 10.1073/pnas.0904837106.PubMedPubMed CentralView ArticleGoogle Scholar
- Wilkins AK, Tidor B, White J, Barton PI: Sensitivity analysis for oscillating dynamical systems. SIAM J Sci Comput. 2009, 31 (4): 2706-2732. 10.1137/070707129.PubMedPubMed CentralView ArticleGoogle Scholar
- Hirota T, Lee JW, St. John PC, Sawa M, Iwaisako K, Noguchi T, Pongsawakul PY, Sonntag T, Welsh DK, Brenner DA, Doyle III FJ, Schultz PG, Kay SA: Identification of small molecule activators of Cryptochrome. Science. 2012, 337 (6098): 1094-1097. 10.1126/science.1223710.PubMedPubMed CentralView ArticleGoogle Scholar
- Ko CH, Takahashi JS: Molecular components of the mammalian circadian clock. Hum Mol Genet. 2006, 15 Spec No 2: R271-R277.PubMedView ArticleGoogle Scholar
- Doyle III FJ, Gunawan R, Bagheri N, Mirsky HP, To TL: Circadian rhythm: A natural, robust, multi-scale control system. Comput Chem Eng. 2006, 30 (10–12): 1700-1711.View ArticleGoogle Scholar
- Herzog ED: Neurons and networks in daily rhythms. Nat Rev Neurosci. 2007, 8 (10): 790-802. 10.1038/nrn2215.PubMedView ArticleGoogle Scholar
- Hughes ME, DiTacchio L, Hayes KR, Vollmers C, Pulivarthy S, Baggs JE, Panda S, Hogenesch JB: Harmonics of circadian gene transcription in mammals. PLoS Genet. 2009, 5 (4): e1000442-10.1371/journal.pgen.1000442.PubMedPubMed CentralView ArticleGoogle Scholar
- Zhang EE, Liu AC, Hirota T, Miraglia LJ, Welch G, Pongsawakul PY, Liu X, Atwood A, Huss JW, Janes J, Su AI, Hogenesch JB, Kay SA: A genome-wide RNAi screen for modifiers of the circadian clock in human cells. Cell. 2009, 139: 199-210. 10.1016/j.cell.2009.08.031.PubMedPubMed CentralView ArticleGoogle Scholar
- Hirota T, Lee JW, Lewis WG, Zhang EE, Breton G, Liu X, Garcia M, Peters EC, Etchegaray JP, Traver D, Schultz PG, Kay SA: High-throughput chemical screen identifies a novel potent modulator of cellular circadian rhythms and reveals CKIα, as a clock regulatory kinase. PLoS Biol. 2010, 8 (12): e1000559-10.1371/journal.pbio.1000559.PubMedPubMed CentralView ArticleGoogle Scholar
- Hirota T, Lewis WG, Liu AC, Lee JW, Schultz PG, Kay SA: A chemical biology approach reveals period shortening of the mammalian circadian clock by specific inhibition of GSK-3beta. Proc Natl Acad Sci USA. 2008, 105 (52): 20746-20751. 10.1073/pnas.0811410106.PubMedPubMed CentralView ArticleGoogle Scholar
- Biegler LT: Nonlinear Programming: Concepts, Algorithms, and Applications to Chemical Processes. 2010, Philidelphia: SIAMView ArticleGoogle Scholar
- Bock HG, Kostina E, Schloder JP: Numerical methods for parameter estimation in nonlinear differential algebraic equations. GAMM-âĂR̆Mitteilungen. 2007, 408 (2): 376-408.View ArticleGoogle Scholar
- Bure EG, Rosenwasser E: The study of the sensitivity of oscillatory systems. Autom Rem Contr. 1974, 7: 1045-1052.Google Scholar
- Kramer M, Rabitz H, Calo J: Sensitivity analysis of oscillatory systems. Appl Math Modell. 1984, 8 (5): 328-350. 10.1016/0307-904X(84)90146-X.View ArticleGoogle Scholar
- Zak DE, Gonye GE, Schwaber JS, Doyle III FJ: Importance of input perturbations and stochastic gene expression in the reverse engineering of genetic regulatory networks: insights from an identifiability analysis of an in silico network. Genome Res. 2003, 13 (11): 2396-2405. 10.1101/gr.1198103.PubMedPubMed CentralView ArticleGoogle Scholar
- Lee C, Etchegaray JP, Cagampang FR, Loudon ASI, Reppert SM: Posttranslational mechanisms regulate the mammalian circadian clock. Cell. 2001, 107 (7): 855-867. 10.1016/S0092-8674(01)00610-9.PubMedView ArticleGoogle Scholar
- Wachter A, Biegler LT: On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math Program. 2006, 106: 25-57. 10.1007/s10107-004-0559-y.View ArticleGoogle Scholar
- Andersson J, Åkesson J, Diehl M: CasADi – A symbolic package for automatic differentiation and optimal control. Recent Advances in Algorithmic Differentiation, Lecture Notes in Computational Science and Engineering. Edited by: Forth S, Hovland P, Phipps E, Utke J, Walther A. 2012, Berlin: SpringerGoogle Scholar
- Hindmarsh AC, Brown PN, Grant KE, Lee SL, Serban R, Shumaker DANE, Woodward CS: SUNDIALS: Suite of nonlinear and differential / algebraic equation solvers. ACM T Math Softw. 2005, 31 (3): 363-396. 10.1145/1089014.1089020.View ArticleGoogle Scholar
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