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PLoS One. 2010; 5(1): e8449.
Published online Jan 5, 2010. doi:  10.1371/journal.pone.0008449
PMCID: PMC2797075

Ligand Depletion in vivo Modulates the Dynamic Range and Cooperativity of Signal Transduction

Vladimir Brezina, Editor

Abstract

Biological signal transduction commonly involves cooperative interactions in the binding of ligands to their receptors. In many cases, ligand concentrations in vivo are close to the value of the dissociation constant of their receptors, resulting in the phenomenon of ligand depletion. Using examples based on rotational bias of bacterial flagellar motors and calcium binding to mammalian calmodulin, we show that ligand depletion diminishes cooperativity and broadens the dynamic range of sensitivity to the signaling ligand. As a result, the same signal transducer responds to different ranges of signal with various degrees of cooperativity according to its effective cellular concentration. Hence, results from in vitro dose-response analyses cannot be applied directly to understand signaling in vivo. Moreover, the receptor concentration is revealed to be a key element in controlling signal transduction and we propose that its modulation constitutes a new way of controlling sensitivity to signals. In addition, through an analysis of the allosteric enzyme aspartate transcarbamylase, we demonstrate that the classical Hill coefficient is not appropriate for characterizing the change in conformational state upon ligand binding to an oligomeric protein (equivalent to a dose-response curve), because it ignores the cooperativity of the conformational change for the corresponding equivalent monomers, which are generally characterized by a Hill coefficient An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e001.jpg. Therefore, we propose a new index of cooperativity based on the comparison of the properties of oligomers and their equivalent monomers.

Introduction

Dose-response is one of the most common experimental approaches used by biologists to monitor the properties of signaling molecules. The power of this approach arises from the fact that the change in any quantifiable physiological response can be measured as a function of the chemical stimulus responsible. In some cases, the resulting curve is sigmoidal, which generally implies cooperative interactions between the binding sites for the ligand that initiates the response (but other explanations are possible — see below). In general, cooperativity (or ultrasensitivity) arises for numerous biological processes regulated by protein-protein or protein-ligand interactions involving multi-site proteins that transduce signals via conformational isomerization [1][3].

Cooperativity has been represented for numerous oligomeric protein systems by the allosteric model of concerted transitions [1]. The model is based on spontaneous transitions between two conformational states, designated An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e002.jpg (for “tense”) and An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e003.jpg (for “relaxed”). The governing principle of the model is that in the absence of any bound ligands, the An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e004.jpg conformation is energetically favored over the An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e005.jpg conformation. However, because the An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e006.jpg conformation has a higher affinity than the An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e007.jpg state for a ligand specific for the protein under consideration, the presence of ligand pulls the An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e008.jpg equilibrium towards the An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e009.jpg state. Under these conditions, a clear distinction can be made between two mathematical functions that describe the behavior of protein-ligand interactions as a function of ligand concentration: 1) the binding function, An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e010.jpg, defined as the fractional occupancy of the ligand binding sites of the protein, taking into account both the An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e011.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e012.jpg states; and 2) the state function, An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e013.jpg, defined as the fraction of molecules in the An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e014.jpg state. The state function An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e015.jpg corresponds closely to what is measured by dose-response analysis for an allosteric “receptor” protein. The definitions of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e016.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e017.jpg, and various related parameters are summarized in Table 1.

Table 1
Summary of terms for cooperativity and ligand depletion.

From its initial application to the sigmoidal oxygen-binding curve of hemoglobin, cooperativity has been conveniently characterized by the Hill coefficient, An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e064.jpg [4], [5]. The value of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e065.jpg is obtained as the slope of the Hill plot: the logarithm of the ratio of occupied to unoccupied binding sites on the ordinate is given as a function of the logarithm of the ligand concentration on the abscissa. The value of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e066.jpg provides an empirical index of cooperativity: its upper limit is the number of interacting sites and its value is directly related to non-cooperative systems, since for a monomeric protein with a single site, An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e067.jpg. The Hill coefficient is widely used, including for dose-response curves, but care must be taken in interpreting its value [6][8], since kinetic effects can alter apparent cooperativity [9] and even a monomeric enzyme can display cooperative behavior, i.e. An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e068.jpg [10], [11].

Cooperativity can also be generated by relatively simple networks [12], for example through competition between two sets of phosphorylation sites [13], as well as sequestration effects involving an inactive complex [14] or more complex signal transduction cascades [15]. The interpretation of values of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e069.jpg, which can be a sign of negative cooperativity [16], also requires careful attention, since even for hemoglobin, binding curves with An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e070.jpg can be generated in the presence of non-stoichiometric concentrations of the positive effector, 2,3-diphosphoglycerate [17].

In addition to cooperativity, the non-linear properties of ultrasensitive systems define a dynamic range of signal intensities for which the responses vary. The greater the degree of cooperativity for a system with respect to signal changes, the narrower the dynamic range over which the response varies. For highly cooperative systems, such as bacterial chemotaxis, elaborate mechanisms have evolved in order to extent the dynamic range of response to changes in the concentrations of attractants or repellants [18], [19].

For all signal transduction systems considered, a predominant effect under physiological conditions is ligand depletion. When the concentrations of receptors are close to the dissociation constant for the relevant ligand, the free concentration of the ligand falls significantly below the total concentration of ligand, which in fact constitutes the actual input signal. This effect can be particularly important under in vivo conditions, for which most protein concentrations and dissociation constants are within the nano- to micro-molar range. The general principle of ligand depletion has been widely recognized [20][22] and various aspects have been considered for biological networks [14], [15], [23]. Here we focus on the consequences of ligand depletion with respect to cooperativity and dynamic range, as visualized for two extreme systems. First, we examine the highly cooperative flagellar motor system [24], [25]. Second, we turn to the minimally cooperative, but ubiquitous example of calmodulin [26], [27] in order to explore the consequences of ligand depletion under diverse conditions that apply in distinct regions of the brain and other organs. Finally, after illustrating why the Hill coefficient is not appropriate for measuring cooperativity of signal transduction, we define a new index of cooperativity, An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e071.jpg, as illustrated with the classical example of the allosteric enzyme aspartate transcarbamylase [28], [29]. We show that An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e072.jpg, based on the introduction of an “equivalent monomer” concept, is a reliable measure of cooperativity for dose-response type curves under all conditions.

Results

Ligand Depletion and Dynamic Range in the Flagellar Motor System

We illustrate the importance of considering ligand depletion with the highly cooperative E. coli flagellar motor system [30], which controls the direction of flagellar rotation in response to the concentration of phosphorylated CheY [31]. The rotational bias of individual motors as a function of CheY-P has been measured using tethered single cells and GFP-CheY [30]. The motor bias reflects a change of rotation from counter-clockwise to clockwise and therefore a change of fractional activation (or state function, An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e073.jpg), which is influenced by the interaction of CheY-P with the 34 units of FliM comprising the motor ring [32]. The data show a high degree of cooperativity, with Hill coefficients of up to 10 reported [30]. In contrast, the fractional occupancy, measured using FRET between CheY and FliM appears to be much less cooperative [31].

For dose-response measurements it is reasonable to assume equivalence to within experimental errors of the concentrations of the free and total ligand only if the protein to which the ligand is bound is present at sufficiently low concentration compared to the dissociation constant. However, for the measurements of the flagella motor system, free and total ligand were determined directly and were found to be far from equivalent [31]. The free concentration is significantly reduced compared to the total concentration, due to binding to FliM, as well as to CheA and CheZ [33]. In order to characterize this effect, we define An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e074.jpg, the response as a function of the total ligand concentration, which is distinct from An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e075.jpg, the response as a function of the free ligand concentration (see Table 1).

When ligand-depletion effects are taken into account, the curve for An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e076.jpg is displaced far to the right of the curve for An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e077.jpg (Figure 1A). In addition, An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e078.jpg is significantly less steep than An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e079.jpg. Moreover, the effect of ligand depletion on response curves is exhibited by all cooperative frameworks based on thermal equilibria, not only strictly concerted-models, such as proposed by Duke et al [34]. Therefore, ligand depletion results in an increase in the dynamic range of signal concentrations sensed by the system, as measured for instance by the differences in total concentration of CheY-P corresponding to An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e080.jpg values between 0.1 and 0.9, which increase from An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e081.jpgµM to An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e082.jpgµM for a full change of response in this range. In comparison, the results presented using An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e083.jpg without taking into account ligand depletion could contribute to an underestimation of the dynamic range, since equivalent response changes would be achieved by increase from An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e084.jpgµM to An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e085.jpgµM. With variations in the concentration of FliM, the dynamic range increases linearly (Figure 1B). More generally for multisite receptors, the dynamic range varies with the number of subunits, as observed for the family of curves in Figure 1B and 1C. Ligand depletion may also account for the discrepancies observed between the results reported by Cluzel et al [30] and other studies [35], [36] showing a much lower apparent cooperativity.

Figure 1
Flagellar motor model.

The Effect of Ligand Depletion on the Response Characteristics of Calmodulin

In contrast to the behavior of a system of high cooperativity as described above, we examined the properties of calmodulin, a key molecule of calcium signaling with relatively low cooperativity [37], for which an analysis based on the MWC model has recently been presented [38]. The protein exists as a small monomer (148 residues), with four distinct calcium binding sites, each characterized by specific dissociation constants for calcium that vary between the low-affinity and high-affinity states [38]. Although the reference ligand binding properties that we used for our analysis are free of ligand-depletion effects [39], we have transformed the data to simulate conditions of ligand depletion, with points that fit the curve for An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e108.jpg (the fractional occupancy as a function of the total calcium concentration) for calmodulin at An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e109.jpgµM (Figure 2A). In addition, we have calculated a series of response curves presented in Figure 2A for the activation of calmodulin by calcium both under conditions with no ligand depletion (An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e110.jpg), as well as under condition with ligand depletion (An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e111.jpg) corresponding to various concentration of calmodulin found in vivo [40]. The differences between An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e112.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e113.jpg are very clear, including a progressive broadening of the dynamic range, with markedly diminished cooperativity as the concentration of calmodulin increases. The corresponding decreases in cooperativity as a function of calmodulin concentration are presented in Figure 2B, showing a dramatic fall off with concentration from the initial value An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e114.jpg under conditions where Ca2+ is in large excess, to cooperativity values for the highest concentrations approaching zero.

Figure 2
Ligand depletion for calmodulin.

Limitations of the Hill Coefficient for Dose-Response Measurements and Introduction of a New Index Applied to Aspartate Transcarbamylase

Since cooperativity of binding is generally evaluated by the Hill coefficient, An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e133.jpg, it is not surprising that the Hill coefficient has also been used to characterize many cooperative biological processes, including the fractional activation of signaling receptors and other proteins. However, as we shall demonstrate here, for conformational isomerization of a multi-site protein, An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e134.jpg is not a reliable measure of cooperativity. In contrast to the cooperativity of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e135.jpg, which varies with the energy difference of the two conformational states, as specified by the conformational isomerization constant, An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e136.jpg, the value of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e137.jpg for An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e138.jpg is independent of the value of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e139.jpg [41], as shown in Figure 3. When conditions of low, intermediate, and high affinity are examined for a hypothetical hexamer (Figure 3, left panels), the corresponding An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e140.jpg curves for cooperativity (Figure 3, middle panels) change appropriately for An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e141.jpg, but are identical for An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e142.jpg in the three cases. As a result, when cooperativity is examined as a function of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e143.jpg (Figure 3, right panels), the point of maximal cooperativity moves to the right for An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e144.jpg of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e145.jpg as affinity decreases, but the maximum value An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e146.jpg for An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e147.jpg displays the opposite pattern.

Figure 3
Dependence of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e148.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e149.jpg and their respective Hill coefficients (An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e150.jpg) on the value of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e151.jpg.

Since An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e180.jpg does not vary with the energetic difference of the two states, the shape of the curves for An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e181.jpg when expressed as Hill plots are invariant for different An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e182.jpg values, as shown in Figure 4. In contrast to the Hill plots of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e183.jpg, for which the shape changes as a function of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e184.jpg values, the curves for An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e185.jpg change only vertical position, not shape. Since cooperativity is generally measured around 50% response, correct results are obtained for An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e186.jpg, but the apparent cooperativity of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e187.jpg at 50%, i.e. An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e188.jpg for a Hill plot, depends on the vertical position of the curve for An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e189.jpg and is only a valid estimate of cooperativity for An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e190.jpg (Figure 4, green curve). The differences in shape between the curves for An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e191.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e192.jpg also explain why the cooperativity curves in Figure 3 (middle panels) tend towards An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e193.jpg at the extremes for An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e194.jpg, but towards An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e195.jpg at the extremes for An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e196.jpg. Values of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e197.jpg are commonly considered to be characteristic of negative cooperativity rather than the absence of cooperativity, but the properties of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e198.jpg curves represent a special case for which the conventional reasoning does not apply. Overall, the analyses presented in Figures 3 and and44 make clear that as a general parameter to characterize An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e199.jpg under any conditions, the Hill coefficient is not a reliable measure of cooperativity.

Figure 4
Hill plots for An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e200.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e201.jpg.

In order to overcome the limitations of the Hill coefficient applied to An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e215.jpg, we reexamined how cooperativity is computed for conformational isomerization using data for the allosteric enzyme aspartate transcarbamylase (ATCase), one of the original examples of allosteric phenomenon [42]. Following the formulation of the two-state MWC model [1], it was recognized that under many conditions, An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e216.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e217.jpg as a function of ligand concentration would not overlap [43]. In a classic study of ATCase, the direct binding of succinate (An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e218.jpg) was compared to the succinate-dependent conformational change (An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e219.jpg) as measured by sedimentation or reactivity of protein sulfydryl groups [44], [45]. ATCase was initially characterized as a tetramer, but later studies revealed a hexamer [46], [47] and subsequent structural studies have thoroughly characterized the two hexameric conformational states, T and R, and their concerted interconversion [48], [49]. Using the parameters of the MWC model established for An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e220.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e221.jpg data on the basis of four sites, the theoretical curves were recalculated with six sites, as presented in Figure 5. Under the experimental conditions employed, the curve for An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e222.jpg is substantially to the left of the curve for An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e223.jpg, which constituted strong evidence a conformational equilibrium pre-existing to ligand binding [45]. When the Hill coefficients are determined at 50% for both the An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e224.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e225.jpg curves, the value of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e226.jpg for An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e227.jpg is a reliable measure of the cooperativity, but the value of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e228.jpg for An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e229.jpg dramatically underestimates the intrinsic cooperativity, as we now demonstrate.

Figure 5
New measure of cooperativity for aspartate transcarbamylase based on an equivalent monomer.

In order to establish the correct intrinsic cooperativity of an oligomeric protein undergoing conformational isomerization, a reference state is required that corresponds to a hypothetical equivalent monomer, characterized by same intrinsic affinities for ligand of the An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e256.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e257.jpg states. A conformational transition of the equivalent monomer as a function of the binding of its ligand can be defined and is represented here by An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e258.jpg, along with the binding to the equivalent monomer represented by An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e259.jpg. For an equivalent monomer, the energy difference between the An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e260.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e261.jpg states is postulated to be An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e262.jpg of the energy for the oligomer, since the energy difference for the oligomer is spread equally over the An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e263.jpg subunits. Therefore, we define An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e264.jpg, a conformational isomerization parameter for the equivalent monomer, where An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e265.jpg (see also Materials and Methods, Eqn 11). When the curves for An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e266.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e267.jpg are compared as in Figure 5A, they cross at the value of 0.5 (which is true for all symmetrical MWC-type systems), but the curve for the equivalent monomer is clearly much more shallow.

With respect to ligand binding, the curves for An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e268.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e269.jpg in Figure 5A differ only slightly and are characterized by Hill coefficients of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e270.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e271.jpg, respectively. In contrast, the An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e272.jpg curve, with a Hill coefficient of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e273.jpg is much less cooperative than the curve for An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e274.jpg, with An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e275.jpg, exactly 6-fold higher than the value for An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e276.jpg. In general, under virtually all conditions An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e277.jpg is characterized by a value of the Hill coefficient, An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e278.jpg (see Figure 6).

Figure 6
Properties of equivalent monomers.

In order to overcome the insensitivity of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e287.jpg for An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e288.jpg to An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e289.jpg (Figure 3) and to rely on an appropriate reference state corresponding to the equivalent monomer, we propose replacing the Hill coefficient for dose-response type behavior by a new cooperativity index, An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e290.jpg (Greek letter nu), based on the ratio of the derivatives of the functions for An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e291.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e292.jpg. The function An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e293.jpg therefore corresponds to the ratio of the slopes exhibited by the responses of the cooperative protein and its equivalent non-cooperative monomer. When the new derivative functions are calculated, for An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e294.jpg for the ATCase data in Figure 5, the values of the derivatives are 0.710 and 0.118, respectively, with a ratio of 6.0. The new cooperativity index An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e295.jpg can also be computed directly from the definition of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e296.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e297.jpg (see Material and Methods, Eqn 15). For ATCase, direct calculation also yields An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e298.jpg.

The revised analysis of ATCase illustrates that the intrinsic cooperativity at An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e299.jpg is always maximal, i.e. equal to the number of binding sites (An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e300.jpg), when compared to the equivalent monomer reference state for symmetrical oligomeric proteins. In other words, for a multi-site protein that undergoes a concerted conformational transition, as defined by the MWC model [1], the maximal cooperativity is always equivalent to the number of ligand-binding sites present and may be grossly underestimated on the basis of the Hill coefficient. This property reflects the absolute linkage, or infinite junctional energy, between binding sites in the MWC framework [34]. When data for the flagellar motor is re-examined in this context, the ratio of the derivatives of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e301.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e302.jpg at 50% (Figure 1) corresponds precisely to the value of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e303.jpg. The value of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e304.jpg represents the intrinsic cooperativity of the protein and An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e305.jpg is not affected by ligand-depletion. For various signal transduction systems, the intrinsic cooperativity can, however, be modulated by ligand depletion effects. In order to characterize the effects of ligand depletion on cooperativity we calculated the effective An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e306.jpg by correcting An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e307.jpg for the ratio of the slopes of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e308.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e309.jpg for corresponding fractional activations. As shown for calmodulin (Figure 2B), as for any sensor protein that possesses intrinsic cooperativity, ligand depletion can dramatically reduce the effective cooperativity in a physiological context. Indeed, this effect can bring the effective cooperativity to near 0 (Figure 2B). Because of non-equivalence of the four calmodulin ligand-binding sites, the non-identical dissociation constants for the sites result in the value of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e310.jpg in Figure 2B.

Discussion

Since many cellular control networks involve cooperative interactions among their components, modeling in the context of complete systems requires accurate estimations of the cooperativity of individual reactions. Since ligand depletion can exert an attenuating effect on cooperativity, it is important to have reliable estimates in the absence of ligand depletion. As illustrated in Figure 3, the Hill coefficient as applied to the state function of the MWC model, An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e311.jpg (equivalent to a dose-response curve) clearly does not reflect the correct cooperativity of the response, due to the invariance in the shape, as visualized in the Hill plot presented in Figure 4. As a result, when applied to the classical allosteric enzyme, aspartate transcarbamylase, the difference between the functions for ligand binding (An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e312.jpg) and change of conformational state (An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e313.jpg) are not meaningfully characterized by their respective Hill coefficients. The value of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e314.jpg for (An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e315.jpg) accurately reflects the correct degree of cooperative binding, since it contrasts with the non-cooperative case, with An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e316.jpg. In comparison, for An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e317.jpg the observed value of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e318.jpg is not meaningful, since the non-cooperative case, as expressed by the corresponding “equivalent monomer,” displays a value of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e319.jpg. The correct extent of cooperativity of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e320.jpg can be calculated from the ratio of these two values, or directly from the new index, An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e321.jpg, as defined in Eqn 15, with An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e322.jpg in the case of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e323.jpg for ATCase.

The results presented here demonstrate that neither dynamic range nor effective cooperativity are properties of sensing proteins that can be considered to be invariant; rather than are likely to vary according to the organ, tissue, or cell-type. The concentrations of most signaling proteins are similar to their dissociation constants, in the nano- to micro-molar range, as for example in the well-characterized compartment of the PSD signaling complex of dendritic spines [50]. For calmodulin, it is particularly clear that ligand-depletion is common under physiological conditions, as shown in Figure 2, with the exact consequences depending on the tissue. Related examples include the interaction of calmodulin with other downstream components, such as calcineurin in the micromolar range [51]. While dose-response curves provide the basic characterizations of “systems” and therefore lie at the core of pharmacological treatments, in the analyses presented here we show that dose-response parameters cannot be reused directly in models of signaling systems. Instead one needs to build “mechanistic” models and run parameter-fitting approaches for particular conditions. Although we emphasized the effects of ligand depletion using the allosteric model [1], the general conclusions would apply equally well to other mechanistic descriptions, including the classical Adair-Klotz formulation [52].

It is also important to emphasize that cooperativity and dynamic range can change with the level of expression of the sensor. It is known that the available pools of signaling proteins can be quickly modified by segregation, inhibition, or change in expression. Because of the extreme cooperativity of the flagellar motor, ligand depletion dramatically increases the dynamic range of the system, as shown in Figure 1, making this system extremely sensitive to concentration effects. Since flagellar protein concentration will ultimate influence these properties, it is therefore clear that by changing the number of motors, bacterial cells could enhance their adaptation properties. Since the number of flagella per bacterial cell can vary considerably [53], this parameter must be taken into account for any complete characterization of chemotaxis [54]. More generally, the use of ligand depletion could be a widespread physiological mechanism for cells to adapt non-linear properties and sensitivity ranges to evolving environmental conditions. Because ligand depletion can decrease the effective cooperativity of transducers in situ and increases the dynamic range, we propose that modifying the concentration of the sensor may be a powerful way to adapt quickly to a new environment and switch from a measurement mode to a detection mode.

As modeling of biological phenomena encompasses systems of increasing complexity, particularly in efforts to develop realistic models of the nervous system [55][59], it is important to represent the underlying molecular processes as accurately as possible. The results presented here, in line with other published findings [14], [15], [20][23], emphasize that cooperativity and its consequences, especially dynamic range, cannot be introduced into models as fixed parameters based on Hill coefficients estimated from in vitro studies. Rather, each set of reaction components must be evaluated separately with respect to effects of concentration in the system examined, in order to describe accurately the functional properties that apply.

Materials and Methods

Dose-Response Relationships for an Oligomeric Protein with Two Conformational States

We consider a multisite signaling protein that can interconvert between two functionally distinct conformational states, a more active state (An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e324.jpg) with a high affinity for ligand (An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e325.jpg) and a less active state (An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e326.jpg), with a low affinity for the ligand. The partition between the two states is characterized by An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e327.jpg, the relative intrinsic stability of the two states in the absence of ligand:

equation image
(1)

The affinities of the An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e329.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e330.jpg states for the specifically bound ligand are characterized by the intrinsic dissociation constants: An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e331.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e332.jpg. For convenience, as originally proposed in the MWC model [1], the ratio of affinities is represented by An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e333.jpg:

equation image
(2)

and the parameter An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e335.jpg is defined as the normalized ligand concentration:

equation image
(3)

Using these parameters [1], for a protein with An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e337.jpg sites, the binding function is given by:

equation image
(4)

and the state function is given by:

equation image
(5)

In order to generalize Eqn 5 to multiple ligands, we introduce a new parameter, An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e340.jpg, to describe the relative stabilization of the T state by a ligand:

equation image
(6)

For a protein with An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e342.jpg sites, at any concentration of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e343.jpg, the state function An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e344.jpg is then given with respect to An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e345.jpg by:

equation image
(7)

For An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e347.jpg different ligands binding on multiple sites to the same protein, An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e348.jpg in the above equation is replaced by the product of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e349.jpg for the respective ligands:

equation image
(8)

Since An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e351.jpg if the number of sites is 0, the concentration of the effector is 0, or the affinities for the An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e352.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e353.jpg states are identical, this formula actually describes the absolute state function, modulated by any possible effector [43].

Calculation of Ligand Depletion

Under conditions of significant ligand depletion, i.e. ligand concentrations in the same range as dissociation constant, the degree of ligand binding to its receptor cannot be calculated directly from the total concentration, because only a fraction of this concentration is “free” and available to participate in the binding equilibrium. For any total concentration, the corresponding free concentration can be calculated with respect to a given receptor concentration as one of the roots of the appropriate second-order equation [22]. However, a simpler approach was used here. We define the parameter An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e354.jpg to define An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e355.jpg as a function of the total concentration. For each value of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e356.jpg, the corresponding value of the total concentration, expressed as An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e357.jpg total, is calculated from the equation:

equation image
(9)

where An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e359.jpg is the concentration of ligand binding sites. Multiplying An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e360.jpg by An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e361.jpg therefore provides a correction factor that when added to An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e362.jpg gives An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e363.jpg.

The Index of Cooperativity, An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e364.jpg, for an Oligomer with Respect to Its Equivalent Monomer

In order to evaluate the cooperativity of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e365.jpg versus An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e366.jpg, it must be compared to the properties of a single-site “equivalent monomer.” For any conditions of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e367.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e368.jpg, and An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e369.jpg, we postulate an equivalent monomer with transitions between monomeric states An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e370.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e371.jpg defined by:

equation image
(10)

where

equation image
(11)

For a symmetrical system composed of identical ligand-binding sites, the fraction of monomers in the An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e374.jpg state is given by:

equation image
(12)

In this case, the curves for An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e376.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e377.jpg as a function of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e378.jpg cross at An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e379.jpg. The slopes of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e380.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e381.jpg versus An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e382.jpg are obtained from, respectively, the following derivatives:

equation image
(13)

and

equation image
(14)

The intrinsic cooperativity or amplification of the signal reflected by the properties of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e385.jpg can then be obtained by a new parameter, represented by the coefficient An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e386.jpg (the Greek letter nu) and calculated from the ratio of the two derivatives above (An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e387.jpg) which simplifies to the equation:

equation image
(15)

The coefficient An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e389.jpg gives the cooperativity of the oligomeric protein for the state function An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e390.jpg in a manner analogous to An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e391.jpg (the Hill coefficient) for the binding function (An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e392.jpg), which describes cooperativity with respect to a monomer that in every case displays a value of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e393.jpg. In contrast, when applied to An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e394.jpg, the Hill coefficient is likely to be substantially less than 1 (see Figure 6B), demonstrating why the Hill coefficient is inappropriate for estimating the cooperativity of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e395.jpg. For a given value of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e396.jpg the lower limit of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e397.jpg is given by An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e398.jpg and the upper limit of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e399.jpg is given by An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e400.jpg, with the curves for An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e401.jpg as a function of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e402.jpg described in Figure 6A. The intersection of the curves for An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e403.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e404.jpg at 0.5 corresponds to the value of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e405.jpg defined as An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e406.jpg and is given by:

equation image
(16)

Under these conditions, An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e408.jpg and the An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e409.jpg cooperativity parameter is at its maximal value: An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e410.jpg (whereas An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e411.jpg for all other values of An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e412.jpg).

Derivation of the Hill Coefficient for an Equivalent Monomer

With respect to ligand binding, compared to Eqn 4 for fractional ligand binding, An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e413.jpg, within the context of the two state MWC model [1], the corresponding equation for fractional binding to the equivalent monomer, An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e414.jpg, is given by:

equation image
(17)

The Hill coefficient, An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e416.jpg, is defined by the derivative:

equation image
(18)

Substituting Eqn 17 for An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e418.jpg yields:

equation image
(19)

In contrast, An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e420.jpg for An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e421.jpg as defined by Eqn 12 yields the derivative:

equation image
(20)

Substituting Eqn 12 for An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e423.jpg yields:

equation image
(21)

Therefore, since An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e425.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e426.jpg, it is clear that:

equation image
(22)

and hence for An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e428.jpg, the Hill coefficient for An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e429.jpg must be An external file that holds a picture, illustration, etc.
Object name is pone.0008449.e430.jpg (additional details in M. Stefan, Thesis, University of Cambridge, 2009).

Acknowledgments

The authors are grateful to Philippe Cluzel and Victor Sourjik for their raw data and enlightening discussions and to Rava A. da Silveira and Nick Goldman and his group for help with the mathematics.

Footnotes

Competing Interests: The authors have declared that no competing interests exist.

Funding: S.E. acknowledges support from the Wellcome Trust. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

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