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Sloppy Models, Parameter Uncertainty, and the Role of
Experimental Design
Joshua F. Apgar1,2,†, David K. Witmer2,3,†, Forest M. White1,4, and Bruce Tidor1,2,3,*
1Department of Biological Engineering
2Computer Science and Artificial Intelligence Laboratory
3Department of Electrical Engineering and Computer Science
4David H. Koch Institute for Integrative Cancer Research, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139, USA
Summary
Computational models are increasingly used to understand and predict complex biological
phenomena. These models contain many unknown parameters, at least some of which are difficult
to measure directly, and instead are estimated by fitting to time-course data. Previous work has
suggested that even with precise data sets, many parameters are unknowable by trajectory
measurements. We examined this question in the context of a pathway model of epidermal growth
factor (EGF) and neuronal growth factor (NGF) signaling. Computationally, we examined a
palette of experimental perturbations that included different doses of EGF and NGF as well as
single and multiple gene knockdowns and overexpressions. While no single experiment could
accurately estimate all of the parameters, experimental design methodology identified a set of five
complementary experiments that could. These results suggest optimism for the prospects for
calibrating even large models, that the success of parameter estimation is intimately linked to the
experimental perturbations used, and that experimental design methodology is important for
parameter fitting of biological models and likely for the accuracy that can be expected for them.
Keywords
epidermal growth factor; model calibration; neuronal growth factor; parameter estimation; sloppy
models
Introduction
One of the goals of systems biology is the construction of computational models that can
accurately predict the response of a biological system to novel stimuli.1-3 Such models serve
to encapsulate our current understanding of biological systems, can indicate gaps in that
understanding, and have the potential to provide a basis for the rational design of
experiments,4,5 clinical interventions,6,7 and synthetic biological systems.8 There are many
varieties of computational models ranging from abstracted data-driven models to highly
detailed molecular-mechanics ones. In this report we focus on the popular class of ordinary
differential equation (ODE) models9-16 typically used to describe systems at the biochemical
and pharmacokinetic level but which are also appropriate at more abstract levels.
*Corresponding Author: Bruce Tidor, MIT Room 32-212, Cambridge MA 02139-4307 USA, Phone: (617) 253-7258, tidor@mit.edu.
†These authors contributed equally to this work
NIH Public Access
Author Manuscript
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Published in final edited form as:
Mol Biosyst. 2010 October ; 6(10): 1890–1900. doi:10.1039/b918098b.
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Constructing an ODE model is comprised of writing kinetic rate equations that describe the
time rate of change of the various chemical species (representing the model topopology),
and determining the unknown parameters in those equations (typically rate constants and
initial concentrations). Unknown parameters are estimated from a variety of data that often
includes time-course measurements of concentration or activity. In this study, we have
focused on the estimation of parameters, which is often referred to as model calibration.
Using computational modeling and experimental design methodology, we have found that
the selection of a set of experiments whose members provide complementary information
can lead to efficient model calibration.
It should be noted that the problem of model calibration is different from model
construction, where increasing numbers of parameters can be used to improve the fit to any
given set of measurements, although parameter uncertainty may remain large. There is a
considerable body of work focused on the problem of model complexity as it relates to
parameter uncertainty.17-20 In general these methods attempt to balance the ability of a more
complex model to reduce fitting errors against the increased likelihood that a more
complicated model will be able to fit the data by chance. Here we fix the model structure
and number of parameters and vary only the measurements taken to develop a strategy for
fitting the constant number of parameters with as little uncertainty as possible.
A detailed treatment of the theory for the current study is present in the Theory section. Here
we provide a framework treatment of that theory. The quality of fit between measurements
and a model can be expressed as the weighted sum-of-squares of the disagreement between
them, which is a chi-squared (χ2) metric. Finding parameter values for a fixed model
topology that minimizes χ2 gives the best-fit parameter values, but because of measurement
uncertainty, different sets of parameter values may be consistent with any given set of
measurements. A common approximation of this parameter uncertainty is to expand χ2 as a
function of the parameters and truncate after the second-order term. The linear term vanishes
because the first derivative of χ2 with respect to parameters is zero when the expansion is
carried out about the best-fit parameter values. χ2 is thus approximated as a constant plus
the second-order terms involving the second derivative of the χ2 quality of fit with respect
to the parameters, known as the Hessian. A given amount of measurement uncertainty leads
to an ellipsoid shaped envelope of constant χ2 in an appropriately scaled parameter space.
Sets of parameters within the envelope are consistent with the measurements and their
associated uncertainty.
Longer axes of the ellipsoid correspond to parameter combinations of greater
uncertainty(i.e., that are less well determined by the measurements), whereas shorter
ellipsoidal axes correspond to parameter combinations of less uncertainty. The mathematics
is such that the aces directions of the ellipsoid are given by the eigenvectors (νi’s) of the
Hessian, and their associated uncertainty is given by the reciprocal of the square root of the
corresponding eigenvalues (λi−1/2). Thus, a set of parameters is well determined by a
collection of measurements when the eigenvalues of the corresponding Hessian are all
sufficiently large that they correspond to small relative parameter uncertainty.
Recently Gutenkunst et al.21 examined parameter uncertainty for 17 models in the EMBL
BioModels Database.22 In their study, the authors assumed noise-free measurements of
every model species sampled continuously in time. The study found that the eigenvalues of
the Hessian spanned a large range (> 106). From this they suggested that, while it may be
possible to estimate some parameters from system-wide data, in practice it would be
difficult or impossible to estimate most of the parameters even from an unrealistically high-
quality data set.21,23,24 Moreover, they pointed out that due to the high eccentricity and
skewness of uncertainly ellipses in parameter space, system-wide data can define system
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behavior better than independent measurements of each parameter and may also produce
better predictions in some circumstances.
Here we extend the previous work by more fully considering the effect of experimental
perturbations on the parameter estimation problem and use experimental design to probe for
particularly effective perturbation experiments. The χ2 goodness of fit metric depends on
both the model and the set of experimental conditions. Some experiments may be more
helpful in calibrating the model than others. In the current work we use effectively
continuous-time data, but many experiments require the selection of discrete time points for
measurements to be taken.25,26 It is well established in the systems biology literature that
optimal experimental design can have an impact on the parameter estimation problem for a
single experiment.23 ,25,27-29 For example, work by Faller et al. has shown for a small model
of a mitogen activated protein kinase (MAPK) cascade that the application of time-varying
stimulation significantly improved the parameter estimation problem.29 Essentially this
corresponds to finding the time-varying input signal that gives the best shaped error
ellipsoid.
In this work, we apply a related approach and examine the extent to which multiple
complementary experiments can be combined to improve the overall parameter estimation
problem. Figure 1B,C shows the result of combining data from two separate experiments.
The parameter estimates from the individual data sets (blue and red ellipses) tightly
constrain one parameter direction and weakly constrain the other. In Figure 1B, the weakly
constrained parameter directions are very similar, so the parameter estimates from the
combined data set are about the same as the estimates from the individual experiments
(green ellipse); by contrast, in Figure 1C the experiments are complementary and together
dramatically constrain the parameter estimates.
Because complementary experiments can constrain parameter estimation space, we have
developed an approach to identify sets of complementary experiments to optimally minimize
parameter uncertainty and tested it in a pathway model of signaling in response to EGF and
NGF.30 We have selected this model so that our results may be directly compared to the
previously published analysis of this model performed by Gutenkunst et al.21 For
consistency, where possible we have used their methods and formalisms. In selecting sets of
complementary experiments, we have explored a palette of candidate experiments consisting
of overexpression or knockdown of single and multiple genes combined with different doses
of EGF and NGF, either alone or in combination.
Computational experimental design methods determined all 48 free parameters to within
10% of their value using just five complementary experiments. Selection of complementary
experiments was essential, as the same level of model calibration could not be achieved with
arbitrary experiments or even with a larger number of “highly informative” experiments.
Moreover, we argue that predictions that are sensitive to information complementary to that
used to parameterize a model could be significantly in error. Experimental design methods
can provide sufficient coverage for all parameter directions and thus guide model calibration
for a given topology to maximize predictive accuracy. As systems biology models are
applied to target identification and clinical trial design, the use of experimental design
approaches to improve model prediction quality could be of crucial importance.
Previous work on the model calibration problem has focused on optimization within the
scope of a single experiment.31,32 Examples include selecting optimal time points,33,34
species,35-38 or stimulus conditions5,39,40 that would be most effective in reducing
parameter uncertainty. However, even highly optimized single experiments are generally
insufficient for model calibration. For this reason, such methods have largely been applied to
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smaller scale problems. The current work is different in spirit in that it addresses the
question of how improved model calibration might result from combinations of experiments
that could collectively define all of the parameters. By design, the individual experiments
may be easier to implement, yet relatively small combinations of simple experiments can
determine all parameters in a medium-sized pathway model.
Theory
In this work we formulate the model calibration problem as a nonlinear least squares
optimization problem, where the goal is to find the set of parameters that minimizes the fit
metric,21
(1)
where nc is the number of experimental conditions, ns is the number of species for which
measurements are available, the indices c and s run over the conditions and species,
respectively, Tc, is the length of the time course for condition c , ys,c(p,t) is the model output
for species s and condition c at time t with parameter set p , ys,c (p*,t) is the corresponding
output for the true model parameterization, and
taken as proportional to the uncertainty of the experimental measurement.
is a weighting factor that is often
There is a significant amount of work devoted to how best to solve this optimization
problem for biological models.41-44 However, in any experimental system, there will always
be uncertainty in the data, which means there will be some range of parameter values that,
while not optimal, cannot be excluded based on the data. Given a maximum acceptable
fitting error, the calibration problem becomes that of finding all parameter sets such
that the error is less than this threshold. In a neighborhood around the optimum
parameterization p *, the least squares cost function can be approximated by its Taylor
series expansion
(2)
Equation 2 describes an np -dimensional ellipsoid in parameter space (np being the number
of fitted parameters), where all of the parameterizations inside the ellipsoid are feasible. The
size and shape of this ellipsoid describe the multidimensional parameter uncertainty. For
example, the longest axis of the ellipsoid corresponds to the parameter direction (that is, the
linear combination of parameters) with the worst error. Likewise, the axis-parallel bounding
box defines the error range for individual parameters. Figure 1 shows an example for a two-
dimensional system. An important distinction illustrated in Figure 1A is that some parameter
directions can have very small uncertainty, while the individual parameters can be quite
uncertain.21 For example, forward and reverse kinetic constants for a binding reaction may
be poorly constrained, yet the equilibrium constant (given by their ratio), can be well
defined.
H, the matrix of second derivatives of the fit metric is known as the Hessian, where Hi, j is
the derivative of χ2(p) with respect to log(pi) and log(pj).
(3)
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The derivative is taken with respect to the natural logarithm of pi to obtain a relative
uncertainty. We can dissect the parameter uncertainty in terms of the eigenvalues λi and
eigenvectors νi of H. The eigenvectors form a natural coordinate system for the ellipsoid,
pointing along the axes. The lengths of the axes are proportional to
smaller eigenvalues correspond to larger relative parameter error.41
, meaning that
Results
Experimental design and computer simulations were applied in tandem to select a collection
of experiments that together could most directly establish each of the rate constant
parameters for the EGF/NGF signaling pathway modeled here. To define all 48 rate
parameters, experimental design procedures must select a set of experiments that together
exercise the model in complementary and sufficiently different ways, rather than simply
choosing multiple different experiments that exercise the model in similar ways. For this
work we chose a palette of experimental perturbations consisting of stimulation with EGF
(107, 105, 103, 10, or 0 molecules/cell-volume) or NGF (4.52×107, 4.52×105, 4.52×103,
45.2, or 0 molecules/cell-volume) individually, or combined treatment with both ligands.
We supplemented this choice of ligand stimulation with a panel of experiments in which
protein expression levels could be modulated by 100-fold overexpression or knockdown for
individual proteins in the network. We then constructed candidate experiments from the
combination of ligand choice and protein expression level changes; specifically, each
experiment was allowed to comprise one stimulation pattern and up to three simultaneous
changes in protein expression level. This experimental set-up resulted in a trial perturbation
set of 164,500 individual computational experiments.
All parameters can be determined to high accuracy
The experiments were evaluated and the number of rate parameter directions determined to
within 10% of their nominal value for each experiment was recorded. The best individual
experiment in this set defined only 29 of the 48 rate parameter directions to this high level of
accuracy. In order to improve on this result, each single experiment was re-evaluated to
determine how many new rate parameter directions could be defined to within 10% of their
nominal value when combined with the best individual experiment. In this manner, a greedy
algorithm was applied to select sequentially sets of experiments based on the ability to
generate tighter bounds on parameter estimates. The results of this greedy algorithm are
shown in Figure 2A. The parameter uncertainties are expressed as an eigenspectrum for each
set of experiments, with increasingly larger experimental sets displayed along the abscissa
and eigenvalues displayed along the ordinate. The horizontal dashed line indicates the 10%
error level, and the number of eigenvalues above the dashed line represents the number of
parameter directions determined to the 10% error level. In Figure 2D, the number of
parameters estimated to the 10% level is shown as a function of the number of experiments
within the experimental set. It is striking to observe that, by properly choosing the correct
combination of experiments, only five total experiments are sufficient to determine all 48
directions (and, indeed, all 48 actual parameters) to within 10% accuracy. This result
indicates that parameter uncertainty, rather than being inherent to biological models, can be
progressively reduced by perturbation experiments.
The five experiments determined here to elucidate all 48 parameters are listed in Table I.
The selected experiments include a tendency for dual stimulation with both EGF and NGF,
combined with multiple protein-expression changes, and a preference for overexpression as
opposed to knockdown of given proteins. Interestingly, the experiments do not appear to
systematically explore all regions of the perturbation space. For instance, four of the five
experiments have a low dose of EGF stimulation, and three of the five experiments have a
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high dose of NGF stimulation. Additionally, NGF stimulation occurs in the absence of EGF
stimulation in one experiment, but EGF stimulation alone is never utilized. The combination
of experiments and the manner in which they explore different aspects of the model in order
to adequately define all parameters is not readily apparent from the chosen experiments.
However, as discussed below, there are some general trends and insights to be gained from
an analysis of the results.
Supplementary Figure S1 shows the location of parameters in the model determined after
each new experiment was added in sequence to the set. The thickness of each reaction arrow
in the Figure indicates whether 0 (thin arrow), 1 (medium arrow), or 2 (thick arrow) of the
parameters associated with the arrow are known to within 10% at that point in the
experimental sequence. Note that parameters closer to the top of the pathway tend to be
determined first, while parameters toward the bottom of the pathway, further from the
application of ligand stimulation, tend to be determined only after multiple experiments are
combined.
The selection of complementary experiments resulting from our experimental design
procedure is non-trivial. For example, if experiments were added sequentially based on their
ability to determine a large number of parameters when applied on their own (Figure 2B) or
if experiments were added randomly (Figure 2C), the performance was much worse; neither
procedure could determine the full 48 parameters with up to 20 experiments, whereas the
greedy experimental design procedure required only five (Figure 2D). Interestingly, the
“random” procedure was about as effective as the “best singles” one beyond the initial few
experiments, which suggests that selecting “good” or “bad” experiments isn’t nearly as
important as choosing complementary ones For example, most of the “best singles”
experiments tended to involve stimulation with EGF and NGF simultaneously
(Supplementary Figure S2), but the type of complementarity required to tease apart all the
parameter directions can be achieved by more subtle variation (Table I). Together these
results show that selection of complementary experiments can very efficiently lead to full
parameterization of complex models and that experimental design procedures are important
to choose an appropriately complementary set.
Relative resolving power of different experiment types
There should be a cost–benefit relationship between the complexity of experimental
perturbations used and the amount of information obtained. Intuitively, we expect it may
take more simple experiments to obtain the same knowledge gleaned from a smaller number
of more complex experiments. The above results included experiments that simultaneously
modified three protein concentrations. To probe whether simpler experiments could achieve
similar results, we repeated the search but limited the experiments to EGF/NGF doses only
(Figure 3A,B), EGF/NGF doses and single knockdowns and overexpressions (Figure 3C,D),
or EGF/NGF doses and up to double knockdowns and overexpressions (Figure 3E,F). For
comparison, results for the full set of experiments with up to triple expression changes are
shown in Figure 3G,H. An experimental set with up to forty experiments was constructed
for each using our greedy experimental design approach, and the maximum number of
parameters determined to within 10% is shown. For each pair of figure panels, the left panel
shows the location in the network of parameters established by the experimental set, with the
same arrow thickness scheme as in Supplementary Figure S1 of thicker arrows indicating
more parameters defined. The eigenvector matrix, a representation of best determined to
least well-determined directions in rate-constant space, is displayed in the right panel for
each sub-section of Figure 3. The eigenvector matrix shows which individual parameters
contribute to each parameter direction. All rate constants could be determined with up to
triple expression changes (as shown above, with five experiments) and 47 of 48 parameters
with up to double expression changes (requiring eight experiments, with no improvement
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from the next 32 most complementary experiments). Most rate constants could be
determined using single changes in protein expression (45 of 48, but requiring 17
experiments), but doses of EGF/NGF alone were only able to establish just over half of the
rate constants (25 of 48, requiring just 2 experiments with no improvement resulting from
the remaining 23 experiments in the class). Thus, more complex perturbation experiments
improved model calibration through establishing a greater number of parameters and
generally doing so with fewer, albeit more difficult, experiments. The tradeoff is such that in
many cases the greater complexity may be justified by the reduced number of total
experiments required. Modifications to future versions of the optimization could be biased
towards reuse of genetic modifications in experiments with different dosage treatments, so
that the greater effort of the former might be better leveraged.
Biochemical basis for complementarity of experiments
Analysis of the five experiments sufficient for determining all the parameters suggests that
one role for some selected experiments is to specifically adjust conditions of enzymatic
reactions so that kcat and KM could be independently determined. Because the calculations
were done in log parameter space, the (+kcat, +KM) subspace direction corresponds to
log(kcat)+log(KM) = log(kcat×KM). Interestingly, the (+kcat, −KM) parameter direction,
corresponding to log(kcat/KM), is easily estimated for many reactions here because in the
wild type many enzymes operate under kcat/KM conditions with substrate concentration well
below KM. Some selected experimental perturbations appear to have been chosen because
they drive substrate concentration sufficiently high as to move outside of kcat/KM conditions
to determine kcat and KM independently (Figure 4).
More stringent parameterization can be achieved with greater experimental effort
It is important to consider whether there are fundamental limits to how accurately biological
models can be parameterized––essentially whether there exist model parameters that are
essentially unknowable. As one step to addressing the question of knowability, we repeated
the optimal experimental design calculations using up to three expression changes, but with
different values of the error threshold. Our previous results were computed with the
requirement that parameters be established to within 10% of their nominal values. The full
set of results is shown in Figure 5. While five experiments were required to establish all 48
parameters to within 10% error, only four experiments were required for the less stringent
37% error. Likewise, as the stringency was increased, greater experimental effort was
required to establish a given number of parameters. Together these results suggest that more
stringent parameterization can generally be achieved with greater experimental effort,
although experimental design procedures may be necessary to define how best to apply
additional effort towards determining new knowledge. Additional experimental complexity
may be necessary to dissect particularly difficult parameter combinations. Whether such
effort is worthwhile may depend on the sensitivity of predictions made by the model to such
difficult parameter combinations.
Discussion
In this study computer simulations and experimental design methods were used to probe the
relationship between experimental perturbations applied to a complex biological system and
the relative certainty with which model parameters describing that system can be
established. These in silico experiments resolve an important question in computational
systems biology. They imply that uncertainty is not inherent to biological network models––
rather, uncertainty can be progressively diminished through sequential addition of
perturbation experiments to a cumulative data set used for model calibration. In the case
studied here, a cellular network activated in response to stimulation by EGF and NGF
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important for cell growth and differentiation, all 48 rate constant parameters could in
principle be fit to within 10% of their nominal value using concentration time courses from
five multi-perturbation experiments. While the accuracy of the measurements and the
desired accuracy of the parameters affect the number of required experiments and their
complexity, all parameters could be estimated to very high accuracy.
An important characteristic of minimal sets of experiments that together define all model
parameters is that the members are mutually complementary. That is, while they may
contain some overlapping information, each experiment should also contain information that
is not provided by the others. The overall parameter uncertainty of a set of experiments is
related to the intersection of the parameter uncertainty associated with each individual
experiment. Thus, the intersection diagrams of Figures 1 and 4 indicate that the directions of
large uncertainty in one experiment should correspond to a direction of smaller uncertainty
in at least one other experiment. This form of complementarity is non-obvious and non-
trivial, but experimental design methodology can efficiently identify sets possessing this
property. In the current example, running enzymes under both kcat/KM and kcat regimes was
one important form of complementarity. Single experiments were generally incapable of
spending sufficient time in each regime, and different experiments were required for each.
The design of single experiments that visit both regimes could be beneficial.
The current case also exhibited a tendency to select protein overexpression experimentsas
opposed to knockdowns, although the generality of this result remains to be seen.
Operationally, in many cases it may be possible and convenient to alter the activity of
proteins with selective inhibitors while leaving the expression level constant. Small
molecule inhibitors or activators also enable time-dependent perturbation, thereby providing
important new degrees of freedom that may permit improved model calibration with fewer
experimental manipulations.5 However, we found certain parameter directions for which
overexpressions were important for their exploration. In the work described here, we used
full trajectories of all concentrations in fitting the model; however, current experimental
technology generally probes only a subset of species and time points. The experimental
design framework described here can be used to determine the most productive species and
time point measurements to make in order to most expeditiously calibrate model parameters.
While a variety of experimental interventions can be used to change protein expression,
another possibility is to make use of natural variation. An examination of the expression
levels of the proteins that correspond to the 19 proteins in the EGF/NGF signaling model as
reported in the GNF SymAtlas database45 is shown in Supplementary Figure S3 and
indicates expression ranges of 100 fold or more across tissues and cell types. Natural
variation on this scale suggests that collecting data in multiple cell types may be an
alternative to using genetic manipulation, but it also suggests that accurate estimation of
parameters and quantification of protein expression levels may be necessary for a model to
be applicable across different cell and tissue types. However, using multiple cell lines
introduces additional complications, as differences in biology and un-modeled effects may
dominate the results. In this case, having a well-calibrated model may make it easier to
distinguish between calibration issues and real biological differences.
For the current work protein knockdowns and overexpressions were treated with no changes
in the expression of other genes, as gene expression is not currently part of the model.30
Expression regulation, when modeled, can readily be account for with the methods
described here.
Many different sets of experiments can be mutually complementary and each can define all
parameters equally well. Furthermore, tradeoffs exist between the complexity of
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experiments available and the number of experiments required to probe fully all parameters.
Multiple genetic changes were particularly effective in the EGF/NGF network studied here,
with a larger number of simpler experiments performing less well overall. Here we designed
sets of experiments to define the parameters of a pre-existing model, as this would allow us
to most directly address the theoretical feasibility of the approach. One result is that rather
complex experiments were called for, and we were curious whether this was a necessity of
the underlying biological system or an artifact of the model itself. We noticed that a few KM
values in the model were quite large, and we built an alternative model with more standard
KM values that fit the experimental as well as the original model (Supplementary Figure S4).
We applied our experimental design methodology but only permitted an experiment to
consist of an EGF/NGF and 100-fold expression change of one protein. All 48 parameters
could be determined using this simpler experimental set in just 15 experiments
(Supplementary Figure S3 and Supplementary Table SI). Interestingly, in this case protein
overexpression experiments dominated underexpression again. Moreover, three genetic
perturbations were re-used, and in two pairs of experiments the same gene was used in both
overexpression and underexpression experiments. Thus, it is likely that numerous, simple
experiments can be used to quantitatively understand and describe even complex biological
systems.
In our analysis we have considered the value of the Hessian only at the optimal parameter
set when computing parameter uncertainty. This corresponds to assuming that the log
parameter errors are Gaussian, an assumption that may not be true for poorly determined
parameter directions. However, as more complementary experiments are added, and as the
parameter uncertainty is reduced, this approximation will become increasingly accurate.46
Throughout out this work, we have stressed the importance of fully determining the
parameters of a model. However, as has been observed by others,21 this may not be
necessary to make a particular prediction. In fact, the corollary to no single experiment
determining all of the parameters is that no single prediction depends on all of the
parameters. This result suggests a potential variation to our method. If a particular model is
intended only to be used in a narrowly defined context where only certain parameters are
sensitive, then the method could choose experiments to define the parameter directions that
spanned the sensitive parameter directions. For example, if a substrate was not to be
expressed at a level above the KM of any of the enzymes that modify it, one could decide to
ignore the parameter errors that pointed in the (+kcat, +KM) direction. However, the results
presented here suggest that the saving in terms of experimental effort may be minimal, as a
small number of experiments was able to cover the entire parameter space, and it comes at
the cost of decreasing the predictive power of the model.
If a complementary set of experiments is used to fully parameterize a model, then there
exists no perturbation to the system for which the parameterization is inadequate. The
identification of therapeutic approaches, drug targets, and treatment regimens essentially
corresponds to identifying perturbations that produce desirable outcomes. To be
meaningfully accurate, such predictions must still be adequately parameterized by the
calibration experiments. The construction of a complementary set of experiments
sequentially reduces all directions of parameter uncertainty. Intuitively, once all the
parameters are well determined, the model will make extremely good predictions whose
error can be bounded through propagation of the remaining uncertainty. (These statements
are predicated on the model topology being sufficiently correct to accurately describe
system dynamics and on the absence of bifurcations and other anomalies within the
remaining parameter uncertainty.) On the other hand, if a model is calibrated with an
incomplete set of experiments, then there exist parameter directions with large uncertainty.
Any experimental prediction whose outcome is sensitive to the undetermined parameter
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directions, such as the effects of drug therapy, for instance, could be grossly incorrect. In
fact, such an experiment might be a poor therapy but an excellent calibration tool. Our
results demonstrate that it is not necessary to modulate the expression level of every protein
in the network in order to determine all parameters and thus fully define the behavior of the
system. Rather, only a complete set of complementary experiments need be used for model
calibration;in principle, excellent predictions should follow.
Methods
The model
In this work we examined a model of EGF and NGF signaling.30,47-51 The model, which can
be found in the BioModels database (BIOMD0000000033),22 consists of 19 distinct
proteins, two extracellular proteins (EGF and NGF, which act as inputs), two cell surface
receptors (EGFR and NGFR), and 15 intracellular proteins. The two receptors and 11 of the
cytoplasmic proteins can be in either an active or inactive state. The remaining four
intracellular proteins are constitutively active, resulting in a total of 32 distinct chemical
species, 26 chemical reactions, and 48 parameters (22 Michaelis constants, KM; 22 catalytic
rate constants, kcat; 2 second-order association constants, kon, and 2 first-order dissociation
constants, koff; note that all initial concentrations are assumed known). Twenty-two of the
reactions are implemented with Michaelis–Menten kinetics, while the remaining four
reactions are the mass-action binding reactions of EGF and NGF to their respective
receptors. One modification was made to the model, which was to put free EGF and NGF
into an extracellular compartment having a volume 1000 times the volume of the
intracellular compartment. This was chosen to better reflect a typical experiment, where the
extracellular volume greatly exceeds the intracellular volume. Preliminary calculations using
a variety of different volumes for the extracellular compartment demonstrated that the
results did not depend strongly on the value for this volume (data not shown).
Objective function
The goal of experimental design for model calibration can be expressed as selecting the
experimental conditions such that the resulting parameter uncertainty ellipsoid will be
prescriptively small. In this work, we focused on maximizing the number of parameters with
uncertainty less than a given threshold. Equation 4 shows this metric FN as functions of the
H
(4)
where nP is the total number of parameter directions and λthresh is determined below. We
chose to include some contribution to our objective function from parameters that did not
meet the threshold to break ties between experiments with the same number of good
parameter directions (thus the term
counts up the number of eigenvalues λi greater than a threshold. By choosing this threshold
to correspond to a 10% relative parameter uncertainty, the function counts up the number of
parameter directions with uncertainty less than 10%.
, rather than zero). Essentially FN(λthresh )
In this work we assume that our data has Gaussian relative error with zero mean and 10%
standard deviation. We weighted concentration differences in Equation 1 by the variance in
the measured value, so σs,c(p*,t)= f × ys.c(p*,t), which makes our least squares estimator a
maximum likelihood estimator. Here f is the relative measurement error, which we set to
0.10 (10%). Equation 1 describes the information from an average data point, where the
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average is over species, conditions, and time. Following the method of Gutenkunst et al.,21
we scaled the cost function by nd. This corresponds to collecting nd independent
measurements of the system, where nd is large enough so that the sample mean approaches
the mean. In this work we assumed 100 times the number of species, to correspond to
discretely sampling the system at 100 time points. Note that as the number of conditions
increase we do not increase the number of data points as this provides useful normalization.
The variance of the parameter uncertainty can be computed by the Cramer–Rao bound.52
The maximum likelihood estimator is asymptotically normal, unbiased, and efficient, so in
the limit of a large number of data points the variance
uncertainty in the ith log parameter eigendirection ψi can be approximated as
of the log parameter
(5)
We define the relative parameter error to be the ratio of the upper and lower values of the
parameter corresponding to the 95% or two standard deviation confidence interval.53
Computing this involves exponentiation of the bounds of log parameter uncertainty.
(6)
As an example,
minimum parameter value is 1.1, which corresponds to ±10% relative error at the 95%
confidence interval. We solved for a λthresh that corresponded to a desired parameter error
by solving Equations 5 and 6; for f = 0.1 and
value, except where noted otherwise.
means that the ratio of the maximum parameter value to the
, λthresh= 0.005.. We used this
Experimental design
Equation 1 shows that the fit metric is a sum over conditions. Likewise, the Hessian of the
fit metric can also be constructed as a sum of the Hessians from each individual condition
(7)
where C is a set of experimental conditions, c indexes individual experimental conditions in
the set, and nc is the number of experiments in C. Thus, we formulated the experimental
design problem as the process of selecting the subset of all possible experiments such that
the sum of the Hessians has the desired properties. Operationally, we simulated each
individual experimental condition in the trial space and computed the Hessian. We then
performed a greedy search to find the best set of nc experiments. At each step of the
algorithm, the Hessian from the previous step was added to the Hessian for each candidate
experiment, and the objective function was evaluated. The trial experiment that led to the
best new Hessian was added to the set. The search terminated if the goal was met or if the
maximum number of experiments was reached. The choice of a greedy search does not
necessarily produce the optimal subset of a given size, indicating that these results are an
upper bound on the number of experiments required to achieve a particular goal.
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Computing Hessians
We computed the individual Hessians of this fit metric for each species in each condition
using the MATLAB SimBiology Toolbox v2.4 (The Mathworks, Natick, MA) following the
method of Gutenkunst et al.21 The ODE system was integrated using ODE15S54 and
sensitivities were computed by complex-step finite differencing.55 The entries in the Hessian
were computed by numerical integration with the trapezoidal rule. Because we evaluated the
Hessian at p = p*, then ys,c(p,t)– ys,c(p*,t) = 0 for all time points, species, and conditions;
thus, the second term in the integrand of Equation 3 (containing the second derivatives of
ys,c) was also zero. For the purpose of this study we dropped this term. Note that even with
imperfect data it may be possible to approximate the Hessian using only first derivatives if
the fit to the data is sufficiently good that the second derivative term is negligible. If this is
not the case then the full Hessian equation should be used, and the rest of the analysis
remains unchanged. We normalized the Hessians as calculated above using f = 0.1, which
represents 10% relative measurement error. The computations were performed in parallel on
128 processors using approximately 800 cpu hours in total.
Supplementary Material
Refer to Web version on PubMed Central for supplementary material.
Acknowledgments
The authors gratefully acknowledge the MIT Computational and Systems Biology community, particularly David
Hagen, Brian Joughin, Doug Lauffenburger, Jacob White, and Dane Wittrup, for stimulating and thoughtful
discussion. This work was partially supported by the National Institutes of Health (U54 CA112967), the MIT
Portugal Program, and the Singapore–MIT Alliance for Research and Technology.
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Figure 1.
(A) Uncertainty ellipse for a simple two-parameter system. The parameters inside the ellipse
are feasible. The major and minor axes of the ellipse are proportional to
respectively. The grey contours are lines of equal objective function value. The bounding
box (dashed magenta lines) shows the single parameter errors. The length and width of the
bounding box is the range of values for the individual parameters. (B) Two non-
complementary experiments. (C) Two complementary experiments. When two experiments
(blue and red ellipses) are combined the resulting parameter estimate (green ellipses) can be
improved as in the case of panel C or not as in the case of panel B depending on the
complementarity of the component experiments.
and ,
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Figure 2.
(A) Search result for sets of experiments that maximize the number of parameters estimated.
Each spectrum is the eigenspectrum of estimation problem. The goal of the design is to
maximize the number of parameters with errors less than 10% (dashed line). (B) Design
based on selecting the best single experiments. (C) Design based on selecting random
experiments. (D) The number of parameters estimated by each search method. By the fifth
experiment the greedy algorithm is able to estimate all 48 parameters to the desired
accuracy. The error bars show the standard deviation for 10000 random searches.
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Figure 3.
The subset of parameters that can be estimated with different levels of experiment. (A,B)
Doses of EGF and NGF alone. (C,D) Doses and single knockouts/overexpressions. (E,F)
Doses and double knockouts/overexpressions. (G,H) Doses and triple knockouts/
overexpressions. The network diagrams show the individual parameter errors. Each arrow
represents a reaction (black for activating, red for inhibitory). Each reaction is parameterized
by two parameters. If both parameters are estimated to 10% the line is thick, if only one of
the two parameters is estimated the line is medium, and if neither parameter is estimated
then the line is thin. The eigenvector matrices pictures on the right show the vector
perspective. The eigenvalues decrease from left to right. The green lines indicates the cutoff
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for 10% relative error. The yellow and red lines indicate directions with 100% and 1000%
error.
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Figure 4.
Experimental perturbations push enzymes from operating in pure
estimation of kcat and KM individually. Experiment #4 calls for the overexpression of the
three enzymes P90Rsk, Akt, and PI3K, which are substrates for the enzymes (A,B) ERK,
(C,D) PI3K, and (E,F) EGFR, respectively. The top row shows the rate of reaction for each
enzyme as a function of its substrate concentration (solid black line), the linearized rate law
conditions to facilitate
at low substrate concentration
experiment #1 (blue shading) and in experiment #4 (red shading). The bottom row shows the
corresponding parameter uncertainty, with KM on the abscissa and kcat on the ordinate.
Uncertainty is shown for experiment #1 alone (blue ellipse), experiment #4 alone (red
ellipse), and the combination of both experiments (green ellipse). In experiment #1 only the
linear region of the enzyme rate curve is explored (blue shading in top row). The linear
(dashed black line), and a histogram of concentration in
regime specifies
corresponds to the blue ellipses in the bottom row of the Figure). The blue ellipses are
extended along the y = x direction, indicating very little uncertainty in log(kcat)–log(KM) but
great uncertainty in log(kcat)+log(KM). In experiment #4 all three substrates are
overexpressed and the saturating portion of the rate curve is populated, where r≈e0kcat,
which specifies the log(kcat) but not log(KM) direction, as indicated by the red ellipses being
aligned nearly along the abscissa. Combining both experiments specifies both rate
parameters, as indicated by the small extent of the green ellipses.
(represented as log(kcat)–log(KM) in log parameter space and
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Figure 5.
Experimental design with different relative parameter errors. The violet line is the original
design with 10% uncertainty. It is interesting to note that with an objective of 37%
uncertainty (gold line) that all but two of the parameters can be estimated with two
experiments, and all parameters can be estimated with four experiments.
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Table I
Parameter-Defining Experimental Set
Exp EGF
(mol./cell)
NGF
(mol./cell)
Overexpressed Knocked Down
1
1.00 × 105
4.56 × 107
Sos, Ras, C3G
2
1.00 × 101
4.56 × 101
Mek, Erk Raf1PPtase
3 0.00
4.56 × 105
BRaf, Rap1 RapGap
4
1.00 × 101
4.56 × 107
P90Rsk, PI3K, Akt
5
1.00 × 103
4.56 × 103
Raf1RasGap
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