Including more microscopic realism in the model requires additional parameters that render the model less identifiable. Note that this sequence of models was constructed in reverse—beginning from a sloppy model, irrelevant parameters were removed one at a time using the manifold boundary approximation method [ 17 , 18 ].
Here we reinterpret this result as a demonstration of the existence of sloppy systems and emphasize the trade-off between mechanistic accuracy and model simplicity. In general, all the models that belong to a sloppy system cannot be ordered in a simple sequence as in Fig 6. There will often be a complex, hierarchical relationship among models similar to that described by the adjacency graphs in reference [ 54 ].
Observations of an EGFR signaling network can be explained by a model that is identifiable and not sloppy. The 18 parameter model has FIM eigenvalues that span fewer than 4 orders of magnitude and are all larger than one.
By including additional mechanisms in the model more parameters the models become increasingly sloppy and less identifiable. The FIM eigenvalues ultimately span more than 16 orders of magnitude, leading to the large parameter uncertainties reported in reference [ 7 ].
In order to account for errors due to ignoring marginal parameters, we need to refine the assumptions underlying Eqs 4 — 6. We adopt a simple hyper-model of the systematic error given by 8 where f is a hyper-parameter that will be estimated from the data, and is another Gaussian random variable with zero mean and standard deviation of one.
We illustrate this concept geometrically in Fig 7. When this ansatz breaks down, it is an indication that relevant mechanisms are missing from the model, i. As in Fig 2 , the model of interest forms a statistical manifold in data space, represented by the black dashed line.
Another more realistic model also forms a statistical manifold of higher dimension red surface. The least squares estimate is the point on the approximate model black dot nearest to the experimental observations. However, the distance from the best fit to the observed data has contributions from both the experimental noise and the model error.
Care must be taken in the interpretation of. We have modeled the systematic error as a random variable. Unlike experimental noise, the size of this uncertainty cannot be decreased by repeated observations. Rather, the stochastic element in the model error represents the unknown approximations in the model. The relevant statistical ensemble is the set of all possible model refinements that could be made to correct the model shortcomings.
We have assumed that the model errors are uncorrelated among data points. We also assume that the model is likely to give worse predictions for data points that also have large experimental variation. These choices are convenient and constitute what is likely the simplest possible such hyper-model. More sophisticated models could be used, and the meta-problem of modeling the error in the model has been addressed in the context of uncertainty quantification [ 44 — 49 ].
In the present context, these assumptions will give us a simple way of estimating the error of the model from data. We will now use our hyper-model to provide a criterion for including additional mechanisms in the model. As a practical matter, the model can be fit using Eq 5 as though there were no approximations in the model.
It is worth noting that the parameter f contains the same information as the likelihood function as is made explicit by Eq Indeed, Eq 12 is a standard statistical formula for estimating parameter uncertainties in ordinary least squares regression in which the scale of the noise is unknown. The standard deviation of the estimate in f is given by 13 We now consider how large an f can be acceptable. This criterion gives 14 as an acceptable value for f.
With this background, we can now revisit the EGFR model above. The experimental conditions of Apgar et al. However, fitting the artificial data typically led to a best fit error greater than , and was never less than 96, Although the parameter estimates remain somewhat constrained, the predictive power of the model is completely lost.
This is because the effective error bars on the data are also larger by a factor of 3. In addition to having a large value for f , the ansatz of Eq 8 breaks down for the fitting the EGFR model.
This is clearly seen by inspecting Fig 5. We speculate that it may be possible to rescue some of the predictive power of the model by implementing a more sophisticated hyper-model, such as introducing a separate f parameter for each time series or including phenomenological parameters to account for correlations in systematic errors.
However, this possibility is beyond the scope of this work, but has been explored in the uncertainty quantification literature [ 44 , 45 , 47 ]. We now consider a model to predict the survival of jejunal crypt clonogens after radiotherapy as in reference [ 55 ].
The LPL and RS models can be combined into a pair of differential equations 15 16 where u t and v t quantify the repairable and non repairable lesions.
The parameter r is the dose rate and is known from experimental conditions. There are a total of six parameters in the composite model, including 3 that are absent from the LPL formulation and 2 that are absent in the RS formulation. Data from reference [ 55 ] explored survival rate for split doses. It reported a statistically significant difference between those cells irradiated initially by a large and then a small dose and vice versa.
The RS model and by extension the composite six parameter model gave reasonable fits to the data; however, the uncertainty in the inferred parameters was quite large. Optimal experimental design was used as an avenue to provide better parameter estimates. In order to identify the optimal experimental conditions for inferring the parameters in Eqs 15 and 16 , the experimental space was first explored numerically.
Four experimental parameters were varied: the radiation level r , the size of each radiation dose, as well as the rest time between doses.
In total, 21, experimental conditions were considered. The FIM for each condition was calculated and nine experimental conditions chosen to augment the original Seven experiments were repeated to confirm results. The result of these 35 experiments and the subsequent fit are given in the supplemental material. Furthermore, the ansatz of Eq 8 is a good approximation for this model and data set.
Because f is not too big less than one and within two standard deviations of zero the model remains predictive in spite of its relative simplicity. Indeed, the model was able to reproduce and explain the asymmetric response to dose size. Similar to the simulated case of the EGFR system, the model is unable to give a reasonable fit to the data. However, the shortcomings of the model are only manifest after the observation conditions have been expanded.
The case of crypt cell survival lacks a comprehensive mechanistic description comparable to what has been done for the EGFR pathway.
From the results of the additional experiments, a simple interpretation of the asymmetry in the first data set is that DNA repair is not monoexponential [ 59 ]. The higher-order terms would contribute disproportionately when intervals are short first data set.
However, it is not known what additional mechanisms should be included to fit the new data. Although beyond the scope of this paper, we speculate that closer inspection of the points of model failure could lead to new insights into DNA repair models.
We further speculate that a hierarchy of models similarly exists for the DNA repair, and our inability to fit the expanded data set indicates that a more complete model would have more than six large eigenvalues in its FIM. Significantly, the predictive power of the model is decreased dramatically when fit to the expanded data set. After including the new data, we found that the model was no longer able to predict the significant asymmetry in the dose response.
This is similar to the EGFR case in which the effective error after fitting the optimal experimental conditions rendered the model non-predictive. Finally, we use the concept of a sloppy system introduced above to explore the limits to which parameters can be accurately estimated and the effect on the predictive power of the model.
There are three cases to consider, corresponding to the three scenarios depicted in Fig 3. The first case arises when there is a clear separation between the relevant physics and the irrelevant mechanisms. In this case, a complete model would have two well-separated groups of eigenvalues, similar to case 1 in Fig 3.
In this case, the irrelevant mechanisms can be safely ignored and the remaining parameters can be estimated to more-or-less arbitrary accuracy. In these cases, the small parameter explicitly suppresses the influence of the irrelevant details.
The ideal gas law, for example, gives very accurate predictions for pressure and volume over a wide range of densities and temperatures without accounting for fluctuations due statistical mechanics considerations.
The second case is when the model ignores several relevant details. If a complete model has large eigenvalues with no analog in an approximate model, then the approximate model will give poor fits to data as we saw for the two test cases above. In this case, the parameter uncertainty is not the bottleneck to model efficacy and is largely irrelevant.
Rather, the systematic errors in the model lead to inaccurate predictions and the model should be refined. Our results suggest that this case may easily occur when optimal experimental design is applied to complex models. Finally, consider the case of a sloppy model for which there is no clear separation between the important and unimportant model details. For many complex systems this appears to be a common occurrence since the eigenvalues are often uniformly spaced on a log scale.
In order for an approximate model to fit the data, there must be a parameter in the model that can be identified with each relevant mechanisms in the true model. Furthermore, in order to have accurate parameter estimates, there should be no small eigenvalues in the approximate model. Therefore, the ideal case is one in which there is a one-to-one correspondence between the large FIM eigenvalues of the complete model and the parameters of the approximate model.
For the ideal scenario in which the parameters of the approximate model correspond to the subspace of largest eigenvalues in the complete model, we can estimate the magnitude of the model error.
Therefore, the cost i. Because eigenvalues in sloppy models are logarithmically spaced, we assume the approximate model is missing eigenvalues from a geometric series with ratio r. In this paper we have explored the relationship among model discrepancy, experimental design, and parameter estimation.
Fig 8 summarizes are primary result. When trying to fit complementary experiments to an approximate model, the best fit may often give an inadequate fit to the data. We explained this result by introducing the concept of sloppy systems as a generalization of sloppy models. Since models are always incomplete, we argued that sloppy models can always be made more accurate by including additional parameters. In addition to making irrelevant parameters identifiable, optimally chosen experiments may often make the ommitted parameters identifiable too, as illustrated in Figs 3 and 4.
We have constructed a simple hyper-model to quantify model error and shown that if a model does not give a good fit then its predictive power is dramatically reduced.
For the two cases considered here, the models are more predictive with unconstrained parameters when fit to a few experiments than they are after fitting to several optimally selected experimental. Parameters inside the ellipses are consistent with the data. Experimental design identifies complementary experiments to minimize the region of consistent parameters.
If the approximate model does not include the this region, the model will be non-predictive for the collection of experiments. It is perhaps surprising that the approximate Michaelis-Menten model is inadequate even if Eq 3 is satisfied. However, one should remember that this condition was derived for a single enzyme-substrate reaction in isolation.
One possible explanation of our results is that approximate Michaelis-Menten kinetics are not valid in a network. This explanation is problematic however, because the approximate Michaelis-Menten model had been used previously to fit real experimental data and make falsifiable predictions for new experiments. Indeed, in spite of its dubious status, the Michaelis-Menten approximation is often used with much success in many systems biology models.
Therefore, while it is true that the Michaelis-Menten approximation is not generally valid, there is considerable evidence that it may sometimes be an safe approximation. To complicate matters, we have forced Eq 3 to be satisfied by requiring K M to be very large. Naively, one would expect this restriction to lead to K m being unidentifiable. While this would also be true for measurements of a single reaction, it does not generalize to the network case, as our results demonstrate.
Furthermore, as the DNA repair results show, our results are not specific to the question of approximate Michaelis-Menten kinetics.
Rather, we have shown that the general question of which physical details are necessary to include in a sloppy model can depend strongly, and in unexpected ways, on which combinations of experiments the model is to explain. A common use for optimal experimental design is model falsification.
We suggested above that errors in the fit could be used to motivate new hypotheses about microscopic mechanisms. This possibility is beyond the scope of the current work that focuses on the implications for parameter estimation in sloppy models. A potential alternative to experimental design for parameter estimation, is experimental design to constrain model predictions [ 28 , 32 ].
Rather than constrain parameter estimates, one seeks to identify a small number of experimental observations that are controlled by the same few parameter combinations as the prediction one would like to make. In this approach, the model parameters remain sloppy, but the model may be predictive in spite of uncertainty about microscopic details.
It may be surprising that a model may be more predictive in the unidentifiable regime than in the identifiable regime. The predictivity of the an unidentifiable model is enabled by the narrow widths of the model manifold in Fig 2. The narrow widths guarantee that even infinite fluctuations in parameters do not correspond to large fluctuations in predictions.
Our results lend some support to this hypothesis; for our test cases removing sloppiness was always accompanied by a decrease in the predictive ability of the model. In previous work, sloppiness has been viewed as a challenge to be overcome or as a disease to be cured [ 5 , 13 , 15 ]. From this perspective the major challenge of sloppy models has been assumed to be the small eigenvalues of the FIM corresponding to practically unidentifiable parameter combinations.
This in turn has led incorrectly to the conflation of sloppiness and practical unidentifiability. As we have argued here, the near uniform spacing of the eigenvalues on a log scale also pose unique challenges for parameter estimation because there is no clear cutoff between relevant and irrelevant mechanisms. In order for an approximate model to be effective, it is important that the microscopic details ommited from the model be irrelevant, i.
When modeling systems for which all the relevant mechanisms are known, the validity of the model can usually be justified by small parameters, e. The small parameter guarantees that the FIM eigenvalues for the irrelevant mechanisms are well-separated from those of the relevant mechanisms e.
Some amount of unidentifiability in the physical system is therefore important for effective modeling. For many complex systems, no such known small parameter exists and sloppy model analysis reveals that there is no sharp distinction between the relevant and irrelevant mechanisms. We speculate that in many cases the system not just the model is intrinsically sloppy because there is no intrinsic scale separation to suppress irrelevant mechanisms in the system. Therefore, a sequence of mechanistically more realistic models would have an eigenvalue structure closer to that in column 1 in Fig 1 rather than Fig 3.
If that is the case, then one should not expect there to exist a mathematical model that can both be accurately calibrated and accurately predict the system behavior. There will always be several parameters that are marginal, i. The chart below shows examples of what models can represent. There are three types of models. Click on the boxes below to learn more about each type of model. There are many advantages to using scientific models.
Click on the icons below to learn more about the advantages of using models. Details—Models cannot include all the details of the objects that they represent. For example, maps cannot include all the details of the features of the earth such as mountains, valleys, etc. Approximations—Most models include some approximations as a convenient way to describe something that happens in nature.
This explanation is problematic however, because the approximate Michaelis-Menten model had been used previously to fit real experimental data and make falsifiable predictions for new experiments. Indeed, in spite of its dubious status, the Michaelis-Menten approximation is often used with much success in many systems biology models. Therefore, while it is true that the Michaelis-Menten approximation is not generally valid, there is considerable evidence that it may sometimes be an safe approximation.
To complicate matters, we have forced Eq 3 to be satisfied by requiring K M to be very large. Naively, one would expect this restriction to lead to K m being unidentifiable. While this would also be true for measurements of a single reaction, it does not generalize to the network case, as our results demonstrate. Furthermore, as the DNA repair results show, our results are not specific to the question of approximate Michaelis-Menten kinetics. Rather, we have shown that the general question of which physical details are necessary to include in a sloppy model can depend strongly, and in unexpected ways, on which combinations of experiments the model is to explain.
A common use for optimal experimental design is model falsification. We suggested above that errors in the fit could be used to motivate new hypotheses about microscopic mechanisms.
This possibility is beyond the scope of the current work that focuses on the implications for parameter estimation in sloppy models. A potential alternative to experimental design for parameter estimation, is experimental design to constrain model predictions [ 28 , 32 ]. Rather than constrain parameter estimates, one seeks to identify a small number of experimental observations that are controlled by the same few parameter combinations as the prediction one would like to make.
In this approach, the model parameters remain sloppy, but the model may be predictive in spite of uncertainty about microscopic details. It may be surprising that a model may be more predictive in the unidentifiable regime than in the identifiable regime. The predictivity of the an unidentifiable model is enabled by the narrow widths of the model manifold in Fig 2. The narrow widths guarantee that even infinite fluctuations in parameters do not correspond to large fluctuations in predictions.
Our results lend some support to this hypothesis; for our test cases removing sloppiness was always accompanied by a decrease in the predictive ability of the model. In previous work, sloppiness has been viewed as a challenge to be overcome or as a disease to be cured [ 5 , 13 , 15 ].
From this perspective the major challenge of sloppy models has been assumed to be the small eigenvalues of the FIM corresponding to practically unidentifiable parameter combinations. This in turn has led incorrectly to the conflation of sloppiness and practical unidentifiability.
As we have argued here, the near uniform spacing of the eigenvalues on a log scale also pose unique challenges for parameter estimation because there is no clear cutoff between relevant and irrelevant mechanisms. In order for an approximate model to be effective, it is important that the microscopic details ommited from the model be irrelevant, i. When modeling systems for which all the relevant mechanisms are known, the validity of the model can usually be justified by small parameters, e.
The small parameter guarantees that the FIM eigenvalues for the irrelevant mechanisms are well-separated from those of the relevant mechanisms e.
Some amount of unidentifiability in the physical system is therefore important for effective modeling. For many complex systems, no such known small parameter exists and sloppy model analysis reveals that there is no sharp distinction between the relevant and irrelevant mechanisms. We speculate that in many cases the system not just the model is intrinsically sloppy because there is no intrinsic scale separation to suppress irrelevant mechanisms in the system.
Therefore, a sequence of mechanistically more realistic models would have an eigenvalue structure closer to that in column 1 in Fig 1 rather than Fig 3. If that is the case, then one should not expect there to exist a mathematical model that can both be accurately calibrated and accurately predict the system behavior.
There will always be several parameters that are marginal, i. In this case there is a fundamental limit to the efficacy of optimal experimental design: attempting to constrain the marginal parameters of a model of a sloppy system reduces the accuracy of the model and limits its predictive ability as we have seen.
Rather than posing a problem for parameter identification in models of complex systems, we argue here that sloppiness is important for successful modeling. Sloppy model analysis reveals that in many cases a behavior of interest is controlled by only a small number of parameter combinations.
This observation has been used to explain why relatively simple models can make useful predictions. Indeed, it has been argued elsewhere that sloppiness may help explain why the world in its microscopic complexity is comprehensible at different scales [ 20 ]. Our results give credence to this position since removing sloppiness from a model reduced its predictive ability.
Another approach is to remove the sloppy parameters from the model. In principle, another simple model may exist whose parameters correspond to the few relevant parameter combinations in the sloppy model.
Parameter estimation in such a model would be relatively straightforward. Recent advances in model reduction suggest that systematic construction of simple models from complex representations may be generally possible [ 17 , 18 , 20 ]. In some branches of physics, the distinction between relevant, irrelevant, and marginal parameters is defined rigorously in terms of the stability of the collective behavior to microscopic variations in mechanistic details as measured by a renormalization group flow.
In that context, relevant parameters correspond to degrees of freedom that must be tuned to achieve a behavior. In this work, we have used relevant and irrelevant less precisely as synonyms for identifiable and unidentifiable as measured by the FIM eigenvalues. This equivalence is reasonable because the the identifiable parameters are those that must be tuned to reproduce a behavior. The equivalence of these definitions was demonstrated in reference [ 12 ].
However, one of the hallmark features of sloppy models is the roughly uniform spacing of FIM eigenvalues, making it difficult to make a clear delineation between relevant and irrelevant parameters. Lacking a clear cutoff between important and unimportant parameter directions means that some physical mechanisms may be either relevant or irrelevant depending on the experimental conditions.
We have shown this explicitly for the two cases considered here. The model of Brown et al. In contrast, it did not contain all of the relevant mechanisms for experimental conditions proposed by Apgar et al.
Similarly, the LPL model is sufficient for modeling single radiation doses, while the RS model is necessary for modeling sequences of varied radiation doses and neither contains all the relevant mechanisms for modeling the experiments described in this work. These results demonstrate the need for a theory of modeling and approximation that identifies which physical mechanisms are relevant for explaining different collective system behaviors.
We have described two approaches that could be the beginnings of such a theory. First, we have introduced the concept of a sloppy system in which multiple models of varying complexity describe the same observations.
Second, we have used a hyper-model to quantify the limitations of a model. Although, most of these ideas have existed in some form in the literature, the unique contribution of this work is synthesizing the concepts to explain why sloppy models pose unique challenges for system identification and why these problems are not shared by unidentifiable models that are not sloppy.
Because simple models are not complete, they cannot make accurate predictions for all experimental conditions. Of course, it is possible to extend a model by including more details in order to extend its range of validity.
In principle, a single, monolithic model could accurately predict the outcome of all possible experiments. This possibility underlies the concept of a sloppy system. Microscopically complete models effectively act as numerical experiments and are a precursor to a more complete theory. In advancing to a more complete understanding of a system, we believe it is useful to consider multiple models of varying complexity and try to understand their limitations.
Simultaneously considering multiple representations creates a rich and insightful theory into the mechanisms driving behavior that allow for abstraction and generalization.
We believe that accounting for the approximations and context of a model are essential for successful modeling. Because many complex systems lack an intrinsic scale separation i. This hypothesis suggests that there is a fundamental limitation of optimal experimental design in sloppy systems due to these marginal parameters; attempting to constrain the marginal parameters of a model of a sloppy system reduces the accuracy of the model and limits its predictive ability.
Mathematical modeling in the face of structural uncertainty is a problem of growing importance across science [ 44 — 49 ]. Because mathematical models by their very nature are not exact replicas of physical processes, it is essential that they include the physical details relevant to the behavior of interest. In some branches of science, most notably several areas in physics, the equations governing some phenomena are sufficiently well-understood that numerical simulations come very near to being surrogates for real experiments.
However, in many areas of complexity science, particularly for systems with fewer symmetries and less homogeneity, which physical details are relevant for explaining a particular behavior remains the theoretical bottleneck. Our results suggest that there is a need for better understanding of and accounting for the approximations in complex models.
In particular, optimal experimental design methods should limit their search space to those experiments for which the model is an accurate approximation. In other words, it is difficult to know a priori which mechanisms are relevant for a particular behavior. We believe that better quantification of uncertainty will enable improved methods of experimental design and the development of accurate models for predicting behavior in complex systems. Animals were maintained in an Association for Assessment and Accreditation of Laboratory Animal Care approved facility, and in accordance with current regulations of the United States Department of Agriculture and Department of Health and Human Services.
The experimental protocol was approved by, and in accordance with, institutional guidelines established by the Institutional Animal Care and Use Committee. We also constructed a similar model using mechanistic mass action kinetics. We first model each chemical reaction as an enzyme and substrate reversibly binding into an enzyme-substrate complex, and then dissociating to yield the original enzyme and the product. This gives four nonlinear ordinary differential equations ODEs for each enzyme substrate ppreaction, including one each for the changes in concentration of the enzyme, the substrate, the enzyme-substrate complex, and the resulting product.
In total, modeling the EGFR network using the same topology as the Brown model by means of mechanistic mass action kinetics requires 54 independent, nonlinear ODES with 70 parameters. These equations are given in the supplement along with an sbml implementation of the mechanistic model and are available on github [ 60 ].
All models were simulated using in-house C, FORTRAN and python routines, including methods to automatically calculate parameter sensitivities, included in the supporting information.
We use the approximate Michaelis-Menten model to simulate the original seven laboratory experiments performed by Brown et al. We then add random noise to results of this simulation and treat the results as if they were actual laboratory results for the experiments. Finally, we fit this data to the mechanistic mass-action model using the geodesic Levenberg-Marquardt algorithm [ 21 , 61 ].
In order to help avoid complications in which the fit is prematurely stuck at manifold boundaries as described in [ 21 ] , all fits were done with regularizing terms to keep parameters from drifting to infinite values. Fits were repeated for many different values of x i 0 i. Weights were varied over four orders of magnitude 0. Using the mechanistic model with parameters from fitting the Brown experiments, we simulate the five experimental conditions proposed by Apgar et al.
We then fit the approximate model to the these data as before. Experimental methods are the same as in reference [ 55 ]. The authors thank Jim Sethna for reading the manuscript and providing helpful feedback.
National Center for Biotechnology Information , U. PLoS Comput Biol. Published online Dec 6. Mason , 3 and Mark K. Howard D. Kathy A. Mark K. Author information Article notes Copyright and License information Disclaimer. Received Feb 16; Accepted Oct This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
This article has been cited by other articles in PMC. Abstract We explore the relationship among experimental design, parameter estimation, and systematic error in sloppy models.
Author Summary Sloppy models are often unidentifiable, i. Introduction Mathematical models play an important role in understanding complex biological systems. Open in a separate window. Fig 1. Sloppiness vs. Fig 2. Model manifold widths define relevant and irrelevant parameters. Fig 3. Experimental design in sloppy system.
Fig 4. Fig 5. Fit of approximate Michaelis-Menten kinetics to mechanistic mass-action data. Quantifying Model Error in Sloppy Systems Motivated by the example of optimal experimental design in EGFR signaling described above, we now propose a simple method to quantify model descrepancy when fitting data to approximate models. Fig 6. An example of a sloppy system. Fig 7. Quantifying model error. DNA repair models We now consider a model to predict the survival of jejunal crypt clonogens after radiotherapy as in reference [ 55 ].
Fundamental limits to parameter estimation in sloppy systems Finally, we use the concept of a sloppy system introduced above to explore the limits to which parameters can be accurately estimated and the effect on the predictive power of the model. Discussion In this paper we have explored the relationship among model discrepancy, experimental design, and parameter estimation. Fig 8. Uncertainty ellipses and approximate models. Mechanistic Mass-Action vs. Approximate Michaelis-Menten Kinetics It is perhaps surprising that the approximate Michaelis-Menten model is inadequate even if Eq 3 is satisfied.
Implications for Optimal Experimental Design A common use for optimal experimental design is model falsification. Parameter Estimation in Sloppy Models In previous work, sloppiness has been viewed as a challenge to be overcome or as a disease to be cured [ 5 , 13 , 15 ].
Relevant and Irrelevant Parameters in Sloppy Systems In some branches of physics, the distinction between relevant, irrelevant, and marginal parameters is defined rigorously in terms of the stability of the collective behavior to microscopic variations in mechanistic details as measured by a renormalization group flow. Radiation Experiments Experimental methods are the same as in reference [ 55 ].
ZIP Click here for additional data file. Acknowledgments The authors thank Jim Sethna for reading the manuscript and providing helpful feedback. Funding Statement The authors received no specific funding for this work. Data Availability All relevant data are within the paper and its Supporting Information files.
References 1. Rosenblueth A, Wiener N. Consequently, chemists continue to use both structural formulas and ball-and-stick models. Ultimately, models are subject to some trade-offs.
You want as much predictive power as possible. At the same time, you also want the model to be as simple as possible. Nature is indifferent to the human need for simplicity and ease of comprehension, however, and many natural phenomena are complex. Just think, for example, about the chain of biochemical processes that take place merely in order to relay information from the photoreceptors in your eye to the visual cortex of your brain.
If you try to incorporate everything that actually happens into a model, it becomes unwieldy and difficult to use. In the end you find that you rely to some degree on approximations and conceptual frameworks that make a process easy to visualize but don't necessarily reflect the true nature of reality.
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