## Abstract

The ability of macromolecules to transduce stimulus information at one site into conformational changes at a distant site, termed “allostery,” is vital for cellular signaling. Here, we propose a link between the sensitivity of allosteric macromolecules to their underlying biophysical parameters, the interrelationships between these parameters, and macromolecular adaptability. We demonstrate that the parameters of a canonical model of the *mSlo* large-conductance Ca^{2+}-activated K^{+} (BK) ion channel are non-identifiable with respect to the equilibrium open probability-voltage relationship, a common functional assay. We construct a reduced model with emergent parameters that are identifiable and expressed as combinations of the original mechanistic parameters. These emergent parameters indicate which coordinated changes in mechanistic parameters can leave assay output unchanged. We predict that these coordinated changes are used by allosteric macromolecules to adapt, and we demonstrate how this prediction can be tested experimentally. We show that these predicted parameter compensations are used in the first reported allosteric phenomena: the Bohr effect, by which hemoglobin adapts to varying pH.

## Introduction

Cellular signaling relies on macromolecules to transduce stimuli information into conformational changes (Changeux and Edelstein, 2005). The mechanisms by which macromolecules accomplish this feat are often allosteric: a small stimulus applied at one area of the macromolecule regulates behavior at locations structurally distant from the active site of stimulation. A detailed and mechanistic understanding of allosteric regulation is a major goal of biophysics (Changeux, 2012, 2013). The Monod-Wyman-Changeux (MWC) model, which provides a physical-chemical interpretation of indirect regulation in terms of the geometry of the regulatory molecule (Monod et al., 1963, 1965; Marzen et al., 2013), has emerged as an essential tool in this effort. Operationally, any given MWC model represents a candidate hypothesis for how allosteric conformational change occurs. If a model is not able to quantitatively fit available data, it is rejected. For models that agree with the data, the model parameter values provide estimates of biophysically meaningful properties that cannot be measured directly. Tremendous effort has gone toward determining which mechanistically relevant parameters best fit available macromolecular data (Colquhoun and Hawkes, 1982, 1995; Horn and Lange, 1983; Blatz and Magleby, 1986; Ball and Sansom, 1989; Kienker, 1989; Ball and Rice, 1992; Colquhoun and Sigworth, 1995; Qin et al., 1996, 2000; Colquhoun et al., 2003; Celentano and Hawkes, 2004; Milescu et al., 2005; Moffatt, 2007). However, it has recently been observed that even simple MWC models suffer from parameter non-identifiability: the data in commonly used activity (or binding) curves do not provide sufficient constraining power to find unique values of the parameters, even if essentially noiseless (Hines et al., 2014; Middendorf and Aldrich, 2017a,b).

Essentially all work on parameter estimation in MWC models has treated non-identifiability as a hurdle to be overcome toward the estimation of individual MWC parameter values, which clearly confer mechanistically meaningful information about the macromolecule under study. Here, we argue that valuable mechanistic information may be lost by focusing on individual parameter values. We demonstrate that non-identifiable datasets admit identifiable “emergent” parameters and argue that these emergent parameters confer mechanistic information about macromolecular function not available from individual parameter values themselves, no matter how accurate.

We begin by studying the causes of parameter non-identifiability in a canonical MWC model of the *mSlo* large-conductance Ca^{2+}-activated K^{+} (BK) ion channel (Horrigan and Aldrich, 2002; Yan and Aldrich, 2010), with respect to two common assays of functional activity. The non-identifiability is shown to arise because the parameters compensate for each other to produce similar model output, rather than because the model contains irrelevant parameters. In particular, we demonstrate that observed parameter non-identifiability is caused by “sloppy” sensitivity of parameters to model output: the model output is highly sensitive to particular nonlinear combinations of parameters and essentially insensitive to others (Brown and Sethna, 2003; Waterfall et al., 2006; Gutenkunst et al., 2007; Transtrum et al., 2010, 2011, 2015).

We address the issue of parameter non-identifiability by constructing “reduced” models of the channel for each of the two functional assays, using the recently developed manifold boundary approximation method (MBAM; Transtrum and Qiu, 2014; Transtrum, 2016 *Preprint*). Each reduced model has fewer parameters than the original model but describe its functional data equally well. Crucially, the parameters of these reduced models are (a) identifiable with respect to the model output and (b) explicitly expressed as emergent combinations of the original MWC parameters. Our reduced models therefore allow for quantitative estimates of biophysically relevant parameters, despite individual parameter non-identifiability.

The emergent parameters of our reduced models indicate which coordinated changes in biophysical parameters may preserve assay output. We therefore interpret our reduced models as predictions encoded in the original MWC model. These predictions confer information about the robustness of a macromolecule’s underlying biophysical parameters with respect to a functional output. We claim that these predictions have physiological and evolutionary relevance, so long as the model is mechanistically relevant and the functional assay adequately captures the macromolecule’s in vivo function.

It is difficult to determine confidently that any given assay or set of assays captures a macromolecule’s in vivo behavior. Here, we choose to study equilibrium binding curves, thereby adopting the hypothesis that the in vivo purpose of a macromolecule is to act as a biological sensor that takes stimuli information as input and produces equilibrium activity as output; this hypothesis is common in information-theoretical studies of MWC models (Tkacik et al., 2008; Martins and Swain, 2011; Olsman and Goentoro, 2016).

Ultimately, the relevance of our predicted parameter compensations, and thereby our assay choice, must be determined experimentally. To this end, we demonstrate that a previous meta-analysis of hemoglobin oxygen-binding curves confirms the physiological relevance of parameter compensations in hemoglobin’s adaptation to varying pH, referred to as the Bohr effect and known to be important for the efficient transportation of oxygen through blood in vivo.

## Materials and methods

The full code used to generate the synthetic data and all analyses is available on GitHub: https://github.com/gbohner/MBAM.

### Synthetic data generation

Our model takes as input *N* voltage and calcium pairs as an array *M* × 1 vector of parameters **θ**. The output of the model is an *N* × 1 vector of open probabilities **θ*** (Fig. 2 B) were chosen to match previous studies (Horrigan and Aldrich, 2002; Miranda et al., 2013).

### Fitting noisy data

Given a **θ** to be estimated, noisy data

We then used the Levenberg-Marquardt solver (Marquardt, 1963) to infer the best-fit parameters

### Computing relative parameter error

Σ* _{i}*, a lower bound on the relative size of the 95% confidence interval for the

*i*-th parameter, was calculated (Gutenkunst et al., 2007; Apgar et al., 2010):

*. For all of our calculations,*

_{i}*N*= 104 and σ = 10%.

### MBAM

MBAM attempts to reduce the number of parameters in a model while quantitatively fitting a given dataset. This is accomplished by reducing the number of parameters in a model one at a time. Each single parameter reduction is the result of eliminating or combining diverging parameters. These diverging parameters are found by searching through parameter space along a trajectory that minimizes the effect on model output. Such a trajectory is found by solving the following system of ordinary differential equations,*t* denotes time spent following the trajectory, **θ*** _{t}* as defined above,

The initial parameter vector **θ**_{0} is taken to be the vector of presumed best-fit parameters **θ**_{0} = **θ***, and the initial direction vector **θ**_{0}) and that we initially move in the direction that goes uphill as little as possible *Preprint*) and Transtrum and Qiu (2014, 2016) for full details.

### Online supplemental material

Our supplemental text contains full algebraic details of the MBAM reductions presented in the paper. Fig. S1 shows RMS cost for each reduced model. Fig. S2 shows how parameter compensations explain the discrepancy between true and inferred values. Fig. S3 shows inferred parameters from the reduced log(*P _{o}*) model.

## Results

The BK channel primarily senses two stimulus signals: membrane voltage and intracellular Ca^{2+} concentration (Horrigan and Aldrich, 2002; Latorre and Brauchi, 2006; Yan and Aldrich, 2010; Miranda et al., 2013). In response to these signals, BK opens its channel gate, allowing potassium ions to permeate. We consider a canonical model of BK gating (Horrigan and Aldrich, 2002; Yan and Aldrich, 2010), shown schematically in Fig. 1 A. The model consists of three functional domains: the channel gate, voltage-sensing domain, and Ca^{2+}-sensing domain. The channel gate is regulated by four identical and independent voltage and Ca^{2+} sensors. Consistent with the MWC framework, each domain can be in one of two conformations: C-O, R-A, ^{2+} subunits, respectively. The function of each domain is defined by an equilibrium constant (*L*, *J*, *K*), and the coupling between domains is mediated by allosteric factors (*C*, *D*, *E*). Formally, the model is given by *J*, *K*, *L*) given by*z _{L}* and

*z*are the partial charges associated with channel opening and voltage sensor activation, respectively. Synthetic data generated from this model is shown in Fig. 1 (B and C).

_{J}The biophysical parameters {*L*_{0}, *z _{L}* ,

*J*

_{0},

*z*,

_{J}*K*,

_{D}*C*,

*D*,

*E*} in Eq. 1 have been estimated with a wide variety of experimental assays. We will focus on the identifiability of these parameters with respect to the steady-state open probabilities of BK at various voltages and Ca

^{2+}concentrations. In the language of MWC models, this “

*P*” curve corresponds to the ubiquitous activity or binding curve. We will also analyze a related common assay (Horrigan and Aldrich, 2002): the base 10 logarithms of the probabilities, log(

_{o}*P*), made possible by single-channel recordings that allow for the determination of very small steady-state open probabilities. We will see that the model exhibits non-identifiability with respect to both assays, but the non-identifiability is more severe for the

_{o}*P*assay. We will demonstrate that this difference in non-identifiability implies that the log(

_{o}*P*) assay may be used to experimentally test for the presence of parameter compensations predicted to hold the

_{o}*P*assay constant.

_{o}### BK parameters are non-identifiable because of sloppiness

We begin by asking the following: how identifiable are the BK model parameters, with respect to each of the *P _{o}* and log(

*P*) assays? Fig. 2 illustrates the problem of non-identifiability via the

_{o}*P*assay: noisy synthetic data generated from the BK model are well fit (within the 10% error bars) by parameter values far from the base values used to generate the data (Fig. 2, A and B). To get a sense of how widely the parameters inferred from noisy

_{o}*P*data may vary, we fit the

_{o}*P*model to 100 noisy synthetic

_{o}*P*datasets (see Materials and methods) generated from the base parameters. We found that the best-fit parameters to noisy data spanned many orders of magnitude; the inferred best-fit parameters for

_{o}*L*,

_{o}*D*are shown in Fig. 2 (C and D). We investigated BK model identifiability systematically by computing lower bounds on the relative parameter error (see Materials and methods) around the base parameter values. Many parameters have significant errors with respect to each assay (Fig. 3 A). The model suffers from non-identifiability.

It has been argued that non-identifiability in scientific models arises because of sloppiness: the model output is extremely sensitive to some combinations of parameters but dramatically insensitive to other combinations (Gutenkunst et al., 2007; Machta et al., 2013). We therefore ask: do either of the *P _{o}*, log(

*P*) assays exhibit sloppiness with respect to the underlying BK model parameters?

_{o}Formally, sloppiness is a feature of a model’s approximate “surface of constant cost.” To understand sloppiness, it is therefore useful to consider a surface of constant cost for a toy two-parameter model (Fig. 3 B). The toy surface is an ellipse that sits in parameter space, centered around the presumed best-fit parameters. In general, the surface of constant cost will have dimension equal to the number of model parameters. Each point on a surface of constant cost identifies parameters whose model output is equivalently different from the output generated by the best-fit parameters, as measured by a cost function. Most commonly, this cost function is the root-mean-squared (RMS) error.

The toy constant cost ellipse illustrates the two key features of sloppiness. (1) One axis of the ellipse is much longer than the other (*w*_{2} >> *w*_{1}); one direction of parameter space therefore constrains model behavior much more than the other. (2) The ellipse is “tilted” rather than aligned with the parameter axes, so each ellipse axis corresponds to a combination of the two parameters. The toy ellipse therefore asserts that some combinations of parameters constrain the model behavior much more than others. The degree to which a parameter combination constrains the model output is encoded in the length of its axis (*w _{i}*). The error bars for the toy parameters θ

_{1}, θ

_{2}are given by Σ

_{1}, Σ

_{2}. Although the model is constrained by the combination of the two parameters in the “stiff” direction of parameter space, neither individual parameter is well constrained.

Because the BK model has many more than two parameters, we cannot easily visualize its ellipsoid of constant cost. We therefore assessed BK model sloppiness by computing a quantity proportional to the lengths of the axes of its constant cost ellipsoids, *P _{o}* assay (Fig. 3 C, red marks) exhibits a greater degree of sloppiness than the log(

*P*) assay (Fig. 3 C, black marks), consistent with the log(

_{o}*P*) assay having more identifiable parameters than the

_{o}*P*assay (Fig. 3 A). Note that each data point in Fig. 3 C corresponds to an axis of the ellipsoid of constant cost, and in general, each axis will correspond to a combination of parameters, as shown in Fig. 3 B. There exists a visible gap (Fig. 3 C, dashed line) between the more sloppy (below line) and more stiff (above line) widths for both assays. Although such a clear gap does not likely exist in general, we will later observe that the minimal number of model reductions needed to produce an identifiable model is equal to the number of axes having widths below this dashed line. This foreshadowing is consistent with the fact that the model reduction procedure leverages the geometry of the model parameter space to remove existing sloppiness.

_{o}### A reduced model for the *P*_{o} assay admits identifiable emergent parameters

_{o}

Having shown that the BK model has non-identifiable parameters because of sloppiness, we now aim to exploit the model’s sloppiness to construct new BK models whose parameters are both identifiable and of mechanistic interest. To construct such models, we use the MBAM (Transtrum and Qiu, 2014; Transtrum, 2016 *Preprint*).

The MBAM algorithm takes three items as input: data, a mathematical model thought to describe the data, and the parameters of the model thought to best fit the data. The goal of the algorithm is to find a point in parameter space having two properties: (1) the model output generated by the parameters must fit the data well; and (2) one or more parameters must be divergent, having values close to 0 or ∞. To find such a point, the algorithm searches through parameter space, along a trajectory that minimizes the effect on model output. The search terminates once divergent parameters are found, and the model is reparameterized such that no parameters equal to 0 or ∞ remain. The algorithm is made to repeat until the newly parameterized models no longer fit the data well (see Materials and methods). The algorithm is deterministic: run on the same data with the same initial conditions, it will return the same reductions.

An overview of our MBAM output with respect to the *P _{o}* assay is shown in Fig. 4. MBAM was run on a set of synthetic data points generated from the BK model and initialized with the “true” BK model parameters that generated these data points. The algorithm was terminated after five iterations, each of which reduced the number of model parameters by one. The first three MBAM iterations produced models that fit the data essentially exactly (Fig. S1). The model produced by the third reduction step (Fig. 4 A) has eliminated five parameters and introduced two new emergent parameters,

To better understand how the reduced models arise, we plotted the numerical values of the model parameters during MBAM searches (Fig. 5, left) and the resulting reduced models (Fig. 5, right). Because MBAM searches along a trajectory of essentially equivalent model behavior, each x-axis “time point” corresponds to a set of parameter values that produce essentially equivalent model output (Fig. 5, left). The algorithm continues searching until one or more parameters diverge to zero or infinity (Fig. 5, red lines). It is interesting to note that as *z _{L}* goes to zero (Fig. 5 A, red line), several nondivergent parameters must compensate (Fig. 5 A, curved black lines) to fit the synthetic data. Subsequent reductions (Fig. 5, B and C) take place without such significant compensation from nondiverging parameters.

The algebraic reparameterizations that give rise to the reduced models in Fig. 5 are worked out in full in the Supplemental text. To understand the emergent parameters biologically, it is instructive to consider the third and “final” reduced model, which is algebraically governed by*J*, *K*) given by

The emergent parameter ϕ_{4} = *CE* represents the effect of [Ca^{2+}] binding on the voltage-sensing process, and *K _{D}* is the binding constant of the ligand,

*J*

_{0}governs the midpoint voltage of the voltage-activation curve, and

*z*controls its slope.

_{J}We next ask: do the reduced models have identifiable parameters? To address this issue, we calculated lower bounds on the relative error for each of the reduced models and found that each reduction step leads to a model with fewer unidentifiable parameters (Fig. 6). It is important to note that although each model reduction step leads to a smaller number of non-identifiable parameters than the one before it, it is not the case that each parameter becomes more identifiable after each reduction step. In the first reduced model, *L*_{0}, *C*, *D*, and *E* and all have much greater relative errors than in the original model (compare Fig. 6 A [table] with Fig. 3 A). Care must therefore be taken in interpreting these intermediate models (Fig. 6, A and B).

In contrast, interpreting the third reduced model (Fig. 6 C) is clear. The model fits the data extremely well (Fig. 4 A) and has parameters that are identifiable and expressed in terms of mechanistically meaningful parameters (Fig. 6 C, table). Variations in underlying MWC parameters that keep these emergent parameters constant may, but do not necessarily, leave assay output unchanged. Conversely, any significant parameter variation that leaves the *P _{o}* assay unchanged must keep the emergent parameters constant. Observe that the differences in base and fit parameters that give rise to the same dataset in Fig. 2 B do indeed keep the emergent parameters essentially constant. Relative to the base parameter, the fit parameter

*C*increased by a factor of 10

^{8}; correspondingly, the parameter

*E*decreased by a factor of 10

^{8}, consistent with constant emergent parameter ϕ

_{4}=

*CE*. Likewise, the value of

^{8}, and correspondingly the value of

*D*increased by a factor of 10

^{8}, consistent with constant emergent parameter

We interpret the emergent parameters as predictions of the original MWC model about potential compensatory mechanisms of the macromolecule. How can we test for the presence of these parameter compensations experimentally? By definition, the compensations are not discernible from the *P _{o}* assay. To address this issue, we therefore turn to the log(

*P*) assay.

_{o}### The log(*P*_{o}) model reduction reveals parameter differences unidentifiable by *P*_{o}

_{o}

_{o}

Having found a reduced, identifiable model for the *P _{o}* assay, we now aim to find such a model for the log(

*P*) assay. Because the log(

_{o}*P*) assay was observed to have fewer “very sloppy” directions than the

_{o}*P*assay (Fig. 3 C), we expected to find an identifiable model after fewer reduction steps than needed for the

_{o}*P*assay. Indeed, MBAM produces an identifiable model after only one reduction step. In this model, each of the original parameters is identifiable except for

_{o}*z*, which has been eliminated (Fig. 7).

_{L}Can we use the reduced log(*P _{o}*) model to detect the presence of large compensatory parameter changes that give rise to identical

*P*datasets? To address this issue, we generated test and base parameters that produced equivalent

_{o}*P*data (up to 10% error), but contained the type of large compensatory changes shown in Fig. 2 B. The test and base parameters were then used to generate noisy synthetic log(

_{o}*P*) data, to which the reduced log(

_{o}*P*) model was fit. Values of the base parameters inferred by the reduced log(

_{o}*P*) model varied many orders of magnitude less than when inferred from noisy

_{o}*P*data generated from the same parameters (compare the black points in Fig. 8 [A and B] with Fig. 2 [C and D]), reflecting the greater identifiably of the reduced log(

_{o}*P*) model.

_{o}This increase in identifiability allows the reduced log(*P _{o}*) model to discern large compensatory parameter changes that leave the

*P*assay constant (Fig. 8, red points). Note that the mean of the inferred parameter values (Fig. 8, solid lines) did not match the true values of the underlying parameters (Fig. 8, dashed lines). This is expected because of compensations from the other parameters (Fig. S2) and does not affect discernibility. Our analysis is limited in that we did not exhaustively sample the parameter space. Rather, we examined several sets of test parameters and found that large compensatory variations from the base parameters were discerned via the reduced log(

_{o}*P*) model in all cases, even when the test parameters were degenerate (Fig. S3). We wish here only to make the point that the log(

_{o}*P*) assay can elucidate parameter variations hidden by the

_{o}*P*assay in practice, not that it must in general.

_{o}## Discussion

Our results demonstrate that reduced models produced by MBAM allow for quantitatively accurate parameter information to be gleaned from models that are non-identifiable with respect to most or all individual parameters. Because the emergent parameters of our reduced models indicate which coordinated changes in parameter values are necessary to preserve the output for a given assay, they may be interpreted as predictions of the MWC model about which biophysical properties can compensate for each other to maintain a functional role. For what purposes would MBAM-elucidated compensatory effects be used by allosteric macromolecules? We propose experiments to test the role of parameter compensation in functional and evolutionary adaptation, both of which have been previously connected to sloppiness (Daniels et al., 2008).

### Emergent parameters may facilitate functional adaptation

It is well known that macromolecules exhibit some amount of functional robustness with respect to experimental perturbations of temperature, pH, salt conditions, and sequence structure (Rennell et al., 1991; Somero, 1995; Suckow et al., 1996; Guo et al., 2004; Weber and Pande, 2012). We propose that this functional robustness may arise from compensatory mechanisms of the type identified here. That is, changes in environment or sequence structure that have little effect on assay output should be observed to have large but compensatory effects in the mechanistic parameters, such that the value of MBAM-identified emergent parameters remain constant. In BK, for example, we may expect that a change in temperature or sequence that marginally effects the *P _{o}* data nonetheless largely increases

*C*but equivalently decreases

*E*.

To test for the presence of a compensatory effect, an experimentalist generally needs at least two assays: a “physiologically relevant” assay whose output should not change upon environmental manipulation, from which compensatory effects may be predicted, and one or more secondary assays, from which the differences in individual parameters can be observed. Obvious candidates for physiologically relevant assays include binding curves (as analyzed here) and physiologically and thermodynamically relevant functions thereof (Wyman, 1967; Di Cera, 1995; Chowdhury and Chanda, 2012). For BK, our results demonstrate that the log(*P _{o}*) assay serves as a good secondary assay for testing compensatory mechanisms predicted from the

*P*assay.

_{o}### Measurements of hemoglobin’s oxygen-binding curve confirms that parameter compensations facilitate functional adaptation

How speculative is this prediction? On one hand, the existence of MBAM-identified parameter compensations is general. Compensations depend only on the underlying sloppiness of the model and assay, which have been demonstrated to be ubiquitous in multiparameter models across systems biology (Gutenkunst et al., 2007). On the other hand, the relevance of MWC parameter compensations in allosteric macromolecules is ultimately a question for experiment. To that end, we note that an extensive meta-analysis of hemoglobin’s oxygen binding confirms the physiological relevance of MWC parameter compensations (Milo et al., 2007).

Milo et al. (2007) analyzed hemoglobin binding curves under varying physiological conditions and in different mammals. To quantitatively study these binding curves, Milo et al. (2007) used an MWC model that assumed the hemoglobin tetramer to be in one of two conformations: relaxed or tense. In each conformation, all four subunits have the same independent affinity for oxygen. The dissociation constants for the relaxed and tense states are *K _{R}*,

*K*, respectively. The equilibrium constant between the fully deoxygenated tense and relaxed states is

_{T}*L*

_{0}. The three mechanistic parameters

*K*,

_{R}*K*,

_{T}*L*

_{0}fully parameterize the model.

The authors found that this three-parameter model was non-identifiable with respect to the saturation curve. Using limit-style arguments that are possible for simple models and essentially equivalent to MBAM (Transtrum and Qiu, 2016), the model was reparameterized to have the emergent, identifiable parameters*L _{T}*

_{0}

_{R}_{4}controls the half-saturation point of oxygen binding

*p*

_{50};

*L*

_{4}controls the cooperativity at this point,

*n*. Moreover, the authors found that

*n*remained constant for human hemoglobin at varying pH (Fig. 9 A).

The results of Milo et al. (2007) therefore present an exceptionally simple framework in which to test our prediction. The *n* assay remains constant upon physiological variation and is controlled by the single emergent parameter *L*_{4}. Is this assay constancy the result of compensations of the underlying MWC parameters, as we predict?

Indeed it is. Across varying pH conditions, *n* stays constant while *p*_{50} varies. Correspondingly, *L*_{4} stays constant as *L _{T}*

_{0}

_{R}_{4}varies (Fig. 9 C). It is immediate from the definitions of

*L*

_{4}and

*L*

_{T}_{0}

_{R}_{4}(Eq. 3) that in order for

*L*

_{4}to remain constant as

*L*

_{T}_{0}

_{R}_{4}varies,

*K*

_{T}^{4}must compensate for changes in

### Emergent parameters may identify evolutionary relevant “neutral spaces”

There is now significant theoretical (Draghi et al., 2010) and experimental (Hayden et al., 2011) evidence, including in allosteric proteins (Raman et al., 2016), that biological systems may evolve by drifting in genotype space such that their primary function remains unchanged. This is known as “neutral evolution” (Wagner, 2005, 2008). Neutral evolution proceeds via mutations that leave current function unchanged but may be advantageous in a subsequent environment. Because the utility of these mutations do not manifest until the correct environment arises, they are said to be cryptic. The spaces of sequences, parameters, or network topologies that give rise to equivalent behaviors are known as neutral spaces (Daniels et al., 2008).

Emergent parameters identified here may define neutral spaces. In particular, cryptic mutations in allosteric macromolecules may give rise to cryptic MWC parameter variations—large but compensatory effects in mechanistic parameters such that the value of MBAM-identified emergent parameters remain constant. This proposal can be tested by searching for cryptic MWC parameter changes between variants of an allosteric macromolecule within and across populations.

At first glance, the study of Milo et al. (2007) provides a way to interrogate this prediction as well. Milo et al. (2007) noted that the assay *p*_{50} stayed relatively constant across 25 different mammals, measured at the same physiological conditions (Fig. 9 B). So too did the emergent parameter that controls it: *L _{T}*

_{0}

_{R}_{4}(Fig. 9 D). This is indeed what we would expect if the emergent parameter identified a neutral space.

Unfortunately, the study by Milo et al. (2007) cannot be used to confirm the presence of a neutral space. If the *L _{T}*

_{0}

_{R}_{4}constancy reflected a neutral space, we would expect

*L*

_{0}and

*K*

_{R}^{4}to vary cryptically between species. Because

*L*

_{0}and

*K*cannot be estimated from the available data, we cannot determine whether

_{R}*p*

_{50}’s constancy across species results from this cryptic variation or from

*L*

_{0}and

*K*

_{R}^{4}each remaining constant. The connection between emergent parameters and neutral spaces remains speculative, though we suspect it may prove an interesting source of future work.

### Conclusions

We have argued that parameter compensation underlies both non-identifiability and adaptability in allosteric macromolecules. We therefore contend that non-identifiability should be expected in any adaptable allosteric macromolecule, so long as the model is mechanistically relevant and the functional assay adequately captures the macromolecule’s function.

Non-identifiability does not itself imply that a model or assay is physiologically relevant. Experimentally verifying that a predicted parameter compensation is actually used, as we have done with hemoglobin and its cooperativity at the half saturation point of oxygen binding, *n*, is paramount. It must also be kept in mind that observed parameter compensations may be spurious rather than meaningful and must be assessed in context. In hemoglobin, the mechanistic relevance of the model parameters and clear physiological relevance of the half saturation point of oxygen binding support the relevance of the compensation we identified.

The enormous success of MWC models in describing allosteric macromolecules presents a wide array of opportunities for further experimentally testing the functional utility of parameter compensations. We anticipate that reduced models will prove essential for extracting mechanistically meaningful quantitative information from MWC models and provide a wealth of readily falsifiable predictions about the emergent properties of allosteric macromolecules.

## Acknowledgments

We would like to thank Miguel Holmgren and Paul Whittredge for discussions and careful reading of previous versions of the manuscript. We thank Daniel Silverman for helpful discussions and Susan Barnes for editorial assistance.

This work was supported by the Gatsby Charitable Foundation.

The authors declare no competing financial interests.

Author contributions: Both authors contributed equally to this work and contributed to all aspects of the work.

Richard W. Aldrich served as editor.

## Footnotes

- Abbreviations used:
- BK
*mSlo*large-conductance Ca^{2+}-activated K^{+}- MBAM
- manifold boundary approximation method
- MWC
- Monod-Wyman-Changeux

- Submitted: 28 December 2016
- Revision received 15 February 2017
- Accepted: 8 March 2017

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