1. Definitions of Mutual Specificity and Mutual Fidelity

As in previous treatments, [22, 25], we consider a network consisting of two pathways, the X pathway and the Y pathway (Fig. 1A). Each pathway has a receptor/signaling component, x₀ and y₀, and a reporter/target component, x₂ and y₂. Notably, the two pathways share a common intermediate component, x₁. Note that one component may be taken to represent the conglomeration of many chemical species. For example x₀ may represent an entire G-protein-coupled receptor complex and several other steps upstream of a shared cascade x₁. Hence, the network shown in Fig. 1A represents the simplest idealized "basic architecture" of a network in which two pathways share components. The input to each pathway will be given by specifying the levels of x₀ and y₀, and the output of each pathway will be measured by x₂ and y₂.

Let us denote the total output of pathway X when the cell is exposed to an × input signal (x₀ > 0, y₀ = 0) as X_out|X_in, read as 'X output given X input', or simply 'X given X'. In a similar fashion we define Y _out |Y _in , the value of y₂ given that the Y pathway is activated (x₀ = 0, y₀ > 0). We also define the crosstalk terms X _out |Y _in (the value of x₂ given that Y is activated) and Y _out |X _in . In this paper, we will use steady-state analysis so as to derive maximal analytical insight, so the outputs defined above refer to steady-state values.

where S _X denotes the output specificity in the X pathway and F _Y denotes the input fidelity in the Y pathway, etc.

MFMS greater than 1 indicates that each of S _X , S _Y , F _X and F _Y are simultaneously (meaning they are evaluated using the same parameters including input levels and connection strengths) greater than 1, and hence the cell signaling network faithfully communicates through both pathways. Note that these definitions may be readily generalized to include more than two intersecting signaling pathways.

In the rest of this paper we explore methods utilized by biological systems to obtain MFMS greater than 1.

2. A Model with Ultrasensitivity

where the Hill exponents, n and m, quantify the degree of ultrasensitivity. For hyperbolic pathways, the Hill exponent is equal to 1, whereas for ultrasensitive pathways, the Hill exponent is greater than 1. Indeed, the greater the Hill exponent, the more switch-like the response. For a Hill number of 1, it an 81-fold change in input to increase output from 10% to 90% maximal. In contrast, for Hill numbers of 2 and 4, it takes a 9-fold and 3-fold change, respectively.

3. Hyperbolic or ultrasensitive signaling pathways can achieve Mutual Specificity

Hence, it is clear that any effort to increase S _X will result in a reciprocal decrease of S _Y , so that both S _X and S _Y cannot be simultaneously greater than one. Neither mutual specificity nor mutual fidelity is possible with the basic architecture and weak activation; thus, some sort of insulating mechanism is required to obtain MFMS for weakly-activated pathways [22].

With a careful selection of parameters mutual specificity of any degree can be obtained (see Additional file 1 section 1a). For ultrasensitive pathways, we have already seen that mutual specificity can be obtained, since hyperbolic pathways are a sub-case of ultrasensitive pathways. So, while both hyperbolic pathways and ultrasensitive pathways can achieve mutual specificity of any degree, ultrasensitive pathways impose less stringent requirements on parameters. For a detailed discussion of the advantages provided by ultrasensitivity see Additional file 1 section 1b.

A pictorial representation of a pathway with no cross-regulation obtaining mutual specificity is given in Fig. 2A. To reiterate, mutual specificity is possible in networks containing hyperbolic or ultrasensitive pathways, even when the topology of such networks is simply the basic architecture without any added insulating mechanism. However, as we show next, it is still impossible to attain mutual fidelity without adding some kind of insulating mechanism to the basic architecture.

4. Mutual Fidelity cannot be obtained by the basic architecture

is impossible.

It should be noted that the specification that this be evaluated at steady state is crucial to this conclusion. There are certain conceivable ways to utilize a time-dependent signal to allow for mutual fidelity and mutual specificity with certain types of activation functions without imposing added regulation.

Note that if we had instead assumed that $x_1^X>x_1^Y$, then we would have concluded that fidelity in the Y pathway is guaranteed, whereas fidelity in the X pathway is impossible. Therefore, we cannot have both X and Y fidelity, i.e. mutual fidelity, regardless of the form of the monotonic activation functions, f ^X and f ^Y . In order to have mutual fidelity, one of the activation functions must be non-monotonic, that is, decreasing somewhere. This cannot be achieved by the basic architecture; it requires some type of added regulation.

5. Insulating mechanisms and cross-regulation

Biological signaling networks that share components are thought to contain one or more insulating mechanisms that provide specificity and fidelity [22, 25, 28–30]. From the analysis above it is clear that insulating mechanisms must be added if the basic architecture is to achieve mutual fidelity and mutual specificity. Here we briefly review three well-known insulating mechanisms, cross-pathway inhibition (CPI), combinatorial signaling (CS) and scaffolding/compartmentalization (SC) [22, 25] (Fig. 1. We will then develop the notion of a cross-regulatory term that facilitates the comparison of different insulating mechanisms. Then, in subsequent sections, we address the effects of ultrasensitivity on the performance of these insulating mechanisms.

Here production of y₂ is inhibited by x₂, with the amount of inhibition depending on the amount of x₂. The parameter ε _g is the IC50 (the inhibitory concentration 50%), which can be interpreted as the amount of x₂ that results in 50% inhibition. When there is no x₂, the production of y₂ is unchanged; when x₂ is much greater than ε _g , y₂ production is nearly completely shut off. Note that this insulating mechanism affects only the Y pathway's output and has no influence on X output. For a discussion of bi-directional mechanisms see Additional file 1 section 3.

Here, R[x₀] represents the combinatorial input. As target x₂ is a coincidence detector, its activity depends on two separate inputs, R and x₁. If either input is zero, then x₂ is also zero. When pathway X is on (and Y off), the coefficient R[x₀] ≡ 1, and signal propagation through the network is identical to the basic architecture. When Y is on, R[x₀] ≡ k_leak, where k_leak, a constant between zero and one, is the normalized basal level of signal flux from Z. Hence, X _out |Y _in will be reduced by a factor of k_leak compared to the basic architecture. Hence, k_leak = 1 has no specificity enhancing effect, whereas k_leak = 0 completely eliminates X output given Y input. As with cross-pathway inhibition, combinatorial signaling only affects one output, in this case the X pathway output.

Signaling scaffolds are proteins that bind to two or more consecutively-acting components of a signaling cascade and, in so doing, facilitate signal transmission between them, (Fig. 1D). A prototypical example is the yeast Ste5 scaffold protein, which binds to all three tiers of the mating MAPK cascade [14]. We refer to this as the sequestering function of scaffolds, to be distinguished from the selective activation function of scaffold proteins [50], which resembles combinatorial signaling [25].

SC becomes increasingly more effective as the exchange parameters D _in , D _out → 0. At this limit, the X and Y pathways have no crosstalk, and hence possess perfect (i.e. infinite) MFMS.

6. Ultrasensitivity can improve insulating mechanism performance

As we have seen, mutual fidelity at steady state is impossible without some kind of additional regulation. In this section we derive maximal values for MFMS for each of the insulating mechanisms, for networks with both linear and ultrasensitive activation. As stated above, in many cases one can show that arbitrarily high degrees of MFMS can be achieved at steady state. Here we derive bounds based on a fixed CRT. We also numerically evaluate the steady state values for each network at different levels of CRT to show that the bounds we derive are in fact sharp.

Linear Activation

For linear activation, deriving expressions for each of the specificity indicators has been done previously [25]. Here we shall re-formulate these in terms of the CRT.

For cross-pathway inhibition, mutual fidelity is not possible; in other words, Y fidelity implies that there is no X fidelity, and vice versa.

$\begin{array}{ll}{F}_X=\frac{x_1^X}{x_1^Y},\hfill & F_Y=\frac{x_1^Y/x_1^X+\left(CRT\right)x_1^Y}{1+\left(CRT\right)x_1^Y}\hfill \end{array}$

Upon inspection F _Y > 1 only when $x_1^Y>x_1^X$, which then makes F _X < 1. Therefore, regardless of the CRT, MFMS ≤ 1.

In the case of combinatorial signaling, however, one can show that $MFMS\le \sqrt{CRT}$. In this case

$MFMS=\min \left\{\left(\frac{\alpha }{\beta }\right),\left(\frac{\beta }{\alpha }\right)CRT,\left(\frac{x_1^X}{x_1^Y}\right)CRT,\left(\frac{x_1^Y}{x_1^X}\right)\right\}.$

The maximum of this expression over all of the parameters occurs when $\alpha /\beta =x_1^Y/x_1^X=\sqrt{CRT}$, and at this point $MFMS=\sqrt{CRT}$.

For scaffolding/compartmentalization (SC), the output specificity and input fidelity readily are calculated:

$\begin{array}{llll}{S}_X=\left(\frac{\alpha }{\beta }\right)\frac{x_1^N|X_{in}}{x_1^C|X_{in}},\hfill & S_Y=\left(\frac{\beta }{\alpha }\right)\frac{x_1^C|Y_{in}}{x_1^N|Y_{in}},\hfill & F_X=\frac{x_1^N|X_{in}}{x_1^N|Y_{in}},\hfill & F_Y=\frac{x_1^C|Y_{in}}{x_1^C|X_{in}}.\hfill \end{array}$

Evaluating these expressions, we find:

$\begin{array}{c}\frac{x_1^N|X_{in}}{x_1^C|X_{in}}=\frac{x_1^C|Y_{in}}{x_1^N|Y_{in}}=1+\frac{d_1}{D}=1+CRT,\\ \begin{array}{ll}\frac{x_1^N|X_{in}}{x_1^N|Y_{in}}=\frac{a_1{x}_0}{b_1{y}_0}(1+CRT),\hfill & \frac{x_1^C|Y_{in}}{x_1^C|X_{in}}=\frac{b_1{y}_0}{a_1{x}_0}(1+CRT)\hfill \end{array}\end{array}$

Hence we obtain MFMS ≤ 1 + CRT.

These bounds, (Fig. 3 dashed lines), are sharp, or the most accurate upper bound, as is apparent from how they are derived. Below we show that the bounds derived for networks with ultrasensitive activation greatly supercede these values.

Ultrasensitivity

In the case of ultrasensitive activation for cross-pathway inhibition (CPI), we can obtain a simple bound on MFMS. Due to the fact that MFMS is the minimum of four quantities the maximum of MFMS is at most as big as the smallest of S _X , S _Y , F _X and F _Y . In the case of CPI it is easiest to bound MFMS by F _Y (In the Additional file 1 we show that this bound for MFMS is sharp: in that there is a choice of parameters so that the MFMS is arbitrarily close to it. See section 2 in the Additional file 1 for derivation).

$F_Y=\frac{Y|Y}{Y|X}=\frac{f^Y(x_1^Y)}{f^Y(x_1^X)}\frac{\left(1+CRT\cdot f^X(x_1^X)\right)}{\left(1+CRT\cdot f^X(x_1^Y)\right)}<\frac{\left(1+CRT\cdot f^X(x_1^X)\right)}{\left(1+CRT\cdot f^X(x_1^Y)\right)}<1+CRT\cdot f^X(x_1^X)<1+CRT$

Therefore we can assert

$MFMS<1+CRT.$

(6.1)

Hence, in contrast to the case with weak-activation and cross-pathway inhibition, where mutual fidelity was impossible, when ultrasensitive, or even hyperbolic, activation is added to this architecture, MFMS > 1 can be obtained.

For combinatorial signaling (CS), the case is much simpler. Regardless of the parameter choice either $x_1^X>x_1^Y$ and therefore $f^Y(x_1^X)>f^Y(x_1^Y)$ and thus

$F_Y=\frac{f^Y(x_1^Y)}{f^Y(x_1^X)}<1,$

or $x_1^Y>x_1^X$ and therefore $f^X(x_1^Y)>f^X(x_1^X)$ and thus

$F_X=\frac{f^X(x_1^X)}{f^X(x_1^Y)}CRT<CRT.$

Thus in any case we have,

$MFMS<CRT$

(6.2)

In both of these cases the degree to which ultrasensitivity helps is hidden. While the bounds for the hyperbolic(n = m = 1) and ultrasensitive case are the same, the speed at which they approach these bounds is much different. With high Hill exponents the constraints on the remaining parameters are much less stringent (see Additional file 1 section 1b). Further high degrees of ultrasensitivity can drastically decrease one of the crossterms X _out |Y _in or Y _out |X _in , see more on this in the next section.

For scaffolding/compartmentalization (SC), making a similar type of bound is less fruitful. Fortunately the exact formula for each of the specificity indicators can be derived straightforwardly. In the case of a symmetric parameter choice, where we let many of the parameters from the X pathway be the same as those in the Y pathway, i.e.

$\begin{array}{llll}{a}_1=b_1\equiv a,\hfill & \alpha =\beta ,\hfill & n=m,\hfill & {\epsilon }_X={\epsilon }_Y=1\hfill \end{array},$

(6.3)

we obtain the bound

$MFMS=\frac{a^n{(1+CRT)}^n+{(1+CRT)}^n{(2+CRT)}^n}{a^n{(1+CRT)}^n+{(2+CRT)}^n}\sim {(1+CRT)}^n.$

(6.4)

In this case, unlike the cases of CPI or CS, the ultrasensitivity and CRT contributions to MFMS are intimately connected. This connection creates a super-linear increase in MFMS due to increasing CRT when ultrasensitivity is greater than one, in contrast with both CPI and CS where MFMS increases only linearly in CRT regardless of the degree of ultrasensitivity. This means that for networks with scaffolding/compartmentalization, even with a low value of the CRT, sufficient ultrasensitivity can serve to greatly increase MFMS, and visa versa. Note that in this symmetric case we have not derived a bound, the MFMS is in fact equal to this value. This is because the symmetric parameter choice greatly simplifies the situation by making S _X = S _Y = F _X = F _Y = MFMS.

Numerical evaluation of the specificity indicators confirm the bounds derived for the networks with ultrasensitive activation are also sharp (data not shown). Further numerical simulation shows that, in the case in which only symmetric parameters are used, as in (6.3), both the maxima and distribution of MFMS values are similar; so the results derived in this case should be representative of the more general case.

The bounds are plotted in Fig. 3 for comparison with those with linear activation. In each case the bounds with ultrasensitive activation clearly supercede those with linear activation. In particular, note the steep increase in MFMS due to the super-linear dependence on CRT in the case of scaffolding/compartmentalization.

To investigate the case where the degree of ultrasensitivity is the limiting factor on MFMS, we numerically evaluated the effect of independently increasing the n and m exponents, while holding the CRT constant and sufficiently high. For each of the insulating mechanisms, MFMS was calculated numerically over a large range of parameters and basic statistics were used. As shown in Fig. 4A and 4B, only n increases MFMS in the case of CPI and only m does this for CS, just as derived in the above bounds. In contrast, for scaffolding/compartmentalization, increasing either n or m increased MFMS. In the case where only one of the Hill exponents is large while the other is kept small the SC network does no better than CPI or CS (data not shown). This is due to the fact that MFMS is a minimum of the four specificity indicators, (1.2), and thus is constrained by the smallest one. However if both n and m are increased simultaneously, MFMS for scaffolding/compartmentalization increases rapidly.

In the case with scaffolding, high degrees of MFMS are achieved at relatively low levels of ultrasensitivity, n or m. For scaffolding with n = m = 10, MFMS is well over 100 (data not shown).

7. Strategies to maximize MFMS using ultrasensitivity

In this section we wish to understand why increasing only one Hill exponent is beneficial for CPI and CS, whereas increasing both is beneficial for scaffolding/compartmentalization as shown in section 6. To explain these observations we study how responsive the specificity and fidelity indicators are to changes in n and m by analyzing their partial derivatives.

which is a necessary and sufficient condition for both n and m to have positive effects on both S _X and S _Y . When this technique is applied to the fidelity indicators we derive the same conditions. Clearly, not all these conditions can be satisfied simultaneously, since either $x_1^X>x_1^Y$ or $x_1^X>x_1^Y$.

After the addition of insulating mechanisms the exact derivatives change slightly, but Eqn. (7.2) still holds for both cross-pathway inhibition (CPI) and combinatorial signaling (CS). In other words adding either of these insulating mechanisms, or both simultaneously (see Additional file 1 section 3) will not change the fact that the equations in (7.1) cannot be simultaneously satisfied.

In CPI, both Y _out |X _in and Y _out |Y _in are decreased due to the inhibition by x₂, which will allow for S _X > 1 and potentially F _Y > 1, but CPI does nothing to decrease the X _out |Y _in term. Thus in order to obtain fidelity in the × pathway, F _X > 1, and hence MFMS > 1 it must be the case that:${\epsilon }_y<x_1^y<{\epsilon }_x<x_1^x$. Under this parameter choice, increasing n has only favorable effects, but increasing m has mixed effects; it increases S _Y but decreases S _X . In the case of CS the relation is the opposite because CS only effects the X _out |Y _in term and has no effect on the others. So the parameters must satisfy the relations: ${\epsilon }_x<x_1^x<{\epsilon }_y<x_1^y$. Again if this were not the case we would not attain MFMS > 1, this time because F _Y < 1. The consequence of this parameter choice, however, is that increasing n decreases S _Y . In both cases the significance of not satisfying one of the above conditions is that increasing m, in the case of CPI, or n, in the case of CS, has detrimental effects on one of the specificity indicators, or increases one of the cross-terms X _out |Y _in or Y _out |X _in . The consequence of equation (7.2) is that networks with either CPI or CS can only utilize ultrasensitivity to decrease one of the crosstalk terms, X _out |Y _in or Y _out |X _in , at a time where the other cross-term must be kept small via cross-regulation. These beneficial effects of ultrasensitivity, however, greatly exceed those in linear or hyperbolic pathways or even those due to cross-regulation. The cross-terms that ultrasensitivity is able to decrease show polynomial decrease (and hence polynomial increase in corresponding specificity indicators) whereas the cross-terms that cross-regulation decrease, in the cases of CS and CPI, show only linear decrease (leading to a linear increase in corresponding specificity indicators). Thus due to the fact that MFMS is a minimum of the four specificity indicators (1.1), the bounds for CPI and CS show only linear increase with CRT.

This allows for the possibility of both n and m to increase MFMS. Therefore with scaffolding/compartmentalization, ultrasensitivity in both the X and Y pathways can simultaneously increase specificity.

Why does SC do so much better? Recall the issue in achieving mutual fidelity: if X _out and Y _out are activated in a monotone way mutual fidelity is impossible. Embellishing the system with insulation mechanisms is a way around this problem and hence insulation mechanisms are responsible for achieving MFMS. However in every case but SC, the problem remains that one of the x₁ steady states must be lower than the other. For this reason there is no way to set the parameters so that increasing the ultrasensitivity simultaneously increases pathway specific variables, X _out |X _in and Y _out |Y _in , while decreasing the crosstalk terms, X _out |Y _in and Y _out |X _in . In SC, because the X pathway is only activated by $x_1^N$ and the Y pathway is only activated by $x_1^C$ one can set the threshold for $x_1^N$ input such that it is above the steady state level when given Y input but below the steady state level when given X input, and visa versa for the $x_1^C$ threshold. The consequences of this are that in this case ultrasensitivity can simultaneously decrease both cross-terms which allows for a polynomial decrease in both terms and thus a polynomial increase in MFMS as a whole, as seen in Figure 3C and equation (6.4). For example for n = 2 SC achieves MFMS at a level an order of magnitude higher than either CPI or CS.

8. Normalization

Specificity in cell signaling pathways is often easy to observe. For instance, yeast cells mate when exposed to mating pheromone and form filaments when starved for nutrients. However this is an observation of whole cell behavior that either happens or not. Quantifying specificity is a more difficult task. Typically one measures the level of pathway specific outputs.

Specificity is defined here as a ratio of two different variables, X and Y, which represent the output of the X and Y pathways, respectively. When measuring this output from a real cell a common thing to measure would be a concentration of an activated kinase or the transcript of a pathway-specific gene; let us call this concentration of gene X. However when comparing this to the output from another pathway, one would be dividing concentration of gene X by the concentration of gene (or kinase) Y. But the concentration at which gene X triggers a physiological endpoint, like mating, may be very different, potentially by orders of magnitude, than the concentration at which gene Y triggers a different output.

To address this issue the two output variables must be normalized somehow so that the construct X _out |X _in is not given in units of concentration of a pathway-specific gene, but given in a unit-less percent of a characteristic concentration for this gene. Two reasonable choices for a characteristic concentration would be the basal level of activation of the gene under no input, or the maximal or steady state level of activation of the gene under its own input. Each choice of a characteristic level should be specific to the system being studied, and hence in the analysis above we assume that such a choice has already been made and the variables are fittingly normalized.

The choice of normalization can have mathematical consequences that lead to a reinterpretation of data. For this reason we discuss the consequences of normalizing by the steady state levels of activation (for a discussion of normalization using basal levels see [25]).

So in this case input fidelity in the X pathway is the same as output specificity in the Y pathway and visa versa. Further the idea of mutual specificity and mutual fidelity are one in the same.

How does this reduction effect the conclusions above? First, clearly mutual specificity is no longer possible without cross regulation, since mutual specificity and fidelity are now equated. The reason that this was possible before and is no longer possible is that attaining MS without cross regulation requires the maximal output of one of the pathways to become large, however with the new normalized species this is impossible as they are both bounded by one.

Secondly the bounds that we derived on MFMS, or more simply specificity in this case, still hold. In each case the degree of MFMS was limited by fidelity, see above. Recall that with no cross regulation mutual specificity of any degree is possible (see section 3) and in fact in this case it is possible to simultaneously maximize S _X , S _Y and F _X but not F _Y (see section 1a of the Additional file 1). Hence the reduction of the problem of achieving MFMS to achieving MF does not make the problem easier.

With a normalization such as this we get a great reduction in the equations to consider. Further the normalized construct makes some intuitive sense. The outputs are given in terms of percent of the activation that occurs when the appropriate input is given. So why not always express the outputs this way? The terms used to normalize the outputs, X _out |Y _in and Y _out |Y _in , are dependent on the input strength. So for a small input the variables are normalized to a smaller number (making them larger) whereas if a large input is used it is the opposite case. Also the input strength of each pathway can be independently varied. So it is possible to choose a strong input for the × pathway and a weak input for the Y pathway or visa versa. These choices could then alter the values of specificity in the system. Again the units of the input to each system is potentially different and hence comparing them is inappropriate. So we are again faced with the problem of normalizing the input based on some characteristic value.

The simple explanation to these issues is that the problem of normalization can be complicated and should be considered on a case-by-case basis. Here we assume that this has been done and all of the variables are unit-less.

Ultrasensitivity can confer output specificity but not input fidelity

To measure specificity we focused on two metrics, mutual output specificity (MS), where both pathways preferentially activate their own outputs, and mutual input fidelity (MF), where both pathways preferentially respond to their own inputs. Examining the network denoted the "basic architecture", a generic, idealized network containing two pathways that share a component, we found that mutual fidelity is impossible at steady-state. Previously we showed that weakly-activated pathways can endow this architecture with neither mutual specificity nor mutual fidelity [22, 25]. Here, we significantly extended this finding by showing that, while both hyperbolic and ultrasensitive pathways can provide mutual specificity, neither can provide mutual fidelity. In fact, our analysis applies not only to hyperbolic and ultrasensitive pathways, but also to any monotonic stimulus-response (input-output) relationship, for reasons discussed below.

Why is input fidelity more difficult to obtain than output specificity? Consider the input-output functions for the X and Y pathways. When both input-output functions are straight lines (as they are when both pathways are weakly activated), they cannot cross; that is, they intersect at the origin but nowhere else. Therefore, it is impossible to choose a pair of input levels so that X output is greater than Y at the lower level of input, and Y output is greater than X at the higher level of input (or visa versa). In other words, mutual specificity is impossible. The input-output functions corresponding to hyperbolic and ultrasensitive pathways are curves, not straight lines. Hence, one can pick parameters such that these curves will cross at some point, and thence it will be possible to pick a pair of input levels on either side of the intersection point that will provide mutual specificity. Mutual fidelity, in contrast, can only be obtained if one of the input-output curves reaches a maximum and then bends back down (see Fig. 2); that is, this curve must be non-monotonic. Non-monotonic behavior in stimulus-response curves generally cannot be achieved by cascades of enzymes that exhibit standard kinetics (even if there is cooperativity or multisite phosphorylation); instead, some sort of negative feedback loop or cross-inhibition will be needed [51]. In this sense, then, mutual input fidelity is more difficult to achieve, and also, perhaps, more difficult to evolve.

Another implication of this result is that, for interconnected networks to exhibit input fidelity (wherein pathways respond preferentially to authentic inputs), the basic network architecture must be embellished with an insulating mechanism(s). This insulating mechanism may work, for instance, by transforming one the stimulus-response curves into a non-monotonic function (e.g. cross-pathway inhibition), or it may act by splitting the stimulus (e.g. scaffolding/compartmentalization) or by splitting a single response curve into two (e.g. combinatorial signaling). Regardless, we should expect insulating mechanisms to be found wherever pathways share components yet exhibit specificity from signal to cellular response.

Ultrasensitivity dramatically improves the performance of insulating mechanisms

Although ultrasensitivity cannot provide mutual fidelity to the basic architecture by itself, it can significantly improve the performance of several different insulating mechanisms. To facilitate comparison, we defined a term denoted the Cross Regulatory Term (CRT) for each of the different insulating mechanisms. In every case this was a non-dimensional term that characterized the "strength" of the insulating mechanism. This allowed us to quantify the degree to which mutual specificity and mutual fidelity (MFMS) increased as a result of increases in the CRT. We found that, as the CRT was increased, there was a much sharper increase in MFMS in those networks featuring ultrasensitive activation than in those with linear (i.e. weak) or hyperbolic activation.

All the networks that we examined displayed sharper increases in MFMS with ultrasensitive activation than with linear or hyperbolic activation. However, for the networks that utilized combinatorial signaling or cross-pathway inhibition, it was not possible to utilize ultrasensitivity simultaneously in both pathways to the benefit of specificity. In other words, increasing the ultrasensitivity in one of the pathways was detrimental. In contrast, networks that combined scaffolding/compartmentalization with ultrasensitive activation could achieve very high levels of MFMS even at low levels of cross regulation, and ultrasensitivity in both pathways was beneficial.

To summarize, the hierarchy we have found is as follows: First, in the absence of an insulating mechanism and in the presence of linear activation, neither mutual specificity nor mutual fidelity is possible. Second, the addition of ultrasensitive activation allows for mutual specificity; in fact, with a careful selection of parameters, S _X , S _Y and F _X can become unbounded. However, mutual fidelity still cannot be achieved. Third, the addition of cross-pathway inhibition or combinatorial signaling to an ultrasensitive system can achieve mutual fidelity, but with a linear dependence on the amount of cross-regulation. Finally, ultrasensitive systems utilizing scaffolding/compartmentalization can realize a super-linear increase in mutual specificity and mutual fidelity as the extent of cross-regulation is increased.

Constraints and opportunities

During evolution, as new signaling pathways emerged from the duplication and divergence of pre-exiting parts, the issue of specificity must have been paramount. Why component sharing is a widespread feature of cellular regulatory networks is a mystery. One possibility is that some low level of crosstalk between pathways is beneficial, but too much is bad; this would explain the existence of both crosstalk and insulating mechanisms. Another possibility is that duplication of part of a pathway, followed by the imposition of an insulating mechanism, is an easier evolutionary path to take than duplication of an entire pathway. Regardless, it seems possible that some of the constraints (e.g. input fidelity is hard to achieve) and opportunities (e.g. ultrasensitivity can help the performance of insulating mechanisms) identified here may have influenced the evolution of signal transduction networks.