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Copyright © 2007 The Biophysical Society. All rights reserved.
Biophysical Journal, Volume 92, Issue 10, 3425-3441, 15 May 2007

doi:10.1529/biophysj.106.090084

Biophysical Theory and Modeling

Mathematical Models of Specificity in Cell Signaling

Lee Bardwell^*, ,  , Xiufen Zou^†, §, Qing Nie^† and Natalia L. Komarova^{†, ‡, ,}  

^* Department of Developmental and Cell Biology, University of California-Irvine, Irvine, California USA
^† Department of Mathematics, University of California-Irvine, Irvine, California USA
^‡ Department of Ecology and Evolutionary Biology, University of California-Irvine, Irvine, California USA
^§ College of Mathematics and Statistics, Wuhan University, Wuhan, China

Address reprint requests to Associate Professor Lee Bardwell, Dept. of Developmental and Cell Biology, 5205 McGaugh Hall, University of California, Irvine, CA 92697-2300. Tel.: 949-824-6902; Fax: 949-824-4709.

Address reprint requests to Associate Professor Natalia Komarova, Dept. of Mathematics, University of California, Irvine, CA 92697-3875. Tel.: 949-824-1268; Fax: 949-824-7993.

The proper growth, development, and survival of an organism requires extensive communication between that organism's cells, and accurate sensing of external conditions. Accordingly, cells sense and respond to a wide variety of chemical and environmental stimuli. Many incoming signals, including hormones such as insulin and adrenalin, are first recognized by a cell surface receptor, and then transmitted to various locations inside the cell by a cascade of signaling proteins that comprise a “signal transduction”, or “signaling”, pathway (Figure 1A). In general, different stimuli trigger distinct cellular responses that are appropriate given the nature of the stimulus. For instance, liver cells respond to insulin by taking up sugar from the blood and storing it as glycogen, whereas they respond to adrenalin by releasing stored sugar into the blood.

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Figure 1

Signaling pathways and networks. (A) A cartoon of a simple signaling cascade. (B) A cartoon of a simple network with a shared component. (C) The yeast mating/invasive growth signaling network, which contains multiple shared components. Mating is initiated by the binding of peptide pheromone to a G-protein-coupled receptor, leading to the activation of a MAP kinase cascade and the induction of mating genes via homodimers of the Ste12 transcription factor. Invasive growth is triggered (in part) by signals transmitted by the Msb2 receptor via the MAP kinase cascade to the Tec1 transcription factor. Mating-pathway-specific components are colored blue, invasive growth-specific components are green, and shared components are yellow. See text for further details.

Different signal transduction pathways are often interconnected to form larger networks. Elements(s) of one pathway may cross-regulate one or more components of another pathway, or multiple distinct pathways may share some of their components (Figure 1B). Such cross-regulation may enable the cell to integrate its overall response when receiving multiple stimuli, and can enable the network to exhibit complex behaviors 1,2,3,4. However, extensive interconnections increase the difficulty of maintaining specificity from signal to cellular response; that is, they increase the likelihood that (under certain circumstances) the activation of one pathway may result in the undesirable activation or inhibition of another pathway. Thus, cross-regulatory interconnections likely evolved hand-in-hand with insulating mechanisms that function to limit undesirable spillover. The problem of “signaling specificity” is to understand the mechanisms that have evolved to maintain specificity from signal to response and to limit “leaking” between pathways, despite extensive interconnections and component sharing 5,6,7,8,9,10,11.

One informative experimental system to study signaling specificity is found in baker's/brewer's yeast (Saccharomyces cerevisiae), where interconnected protein kinase cascades regulate two distinct biological endpoints: mating and filamentous invasive growth 12. These endpoints are triggered by distinct stimuli, leading to the differential activation of downstream mitogen-activated protein kinases (MAPKs), and induction of an appropriate set of target genes (Figure 1C). Mating is initiated by mating pheromone and results in induction of genes mediating the fusion of two haploid cells 13,14, whereas invasive growth is triggered by mechanical and nutrient cues and results in changes in cell shape and adhesiveness 15. Both pathways use the sequentially acting protein kinases Ste20^MAP4K, Ste11^MAP3K, and Ste7^MAP2K16,17. However, mating pheromone stimulates the activation of both Fus3^MAPK and Kss1^MAPK, whereas only Kss1 is activated during invasive growth 18,19. Despite this extensive component sharing, mating and invasive growth are normally reasonably well insulated from one another: cells exposed to mating pheromone do not initiate invasive growth, and cells growing invasively do not induce the mating program. However, mutations in certain key components of this signaling network can compromise specificity, so that treatment with pheromone leads to the induction of invasive growth genes, for example, 18,20,21,22,23. These findings, together with observations from other experimental systems, demonstrate that cells have evolved mechanisms that promote signaling specificity by limiting the extent of leaking between pathways that share similar or identical components 7,8,12.

Because disruption of signaling specificity may play a role in the pathogenesis of cancer and other diseases 5,24,25,26,27, further understanding of mechanisms that promote specificity is warranted, both at the experimental and theoretical level. Here we extend our recently developed framework for the analysis of networks containing two or more signaling pathways 28. We concern ourselves with the following questions: How can the concept of specificity be precisely defined? Are there fundamental limits to specificity imposed by certain network architectures? How effective, in theory, are some of the insulating mechanisms found in nature that have been proposed to enhance specificity? Do certain insulating mechanisms impose additional constraints on the network? What is the effect of noise in stochastic signaling networks on specificity and fidelity? Finally, are there common features or emergent properties of signaling networks that exhibit a reasonable degree of specificity despite undesired signal crossover?

This article is organized as follows: first we describe a theoretical framework that allows us to reason about cell signaling and quantify signal specificity; we also give a mathematical description of very simple signaling networks and examine their specificity properties. Next we discuss several different insulating mechanisms, including combinatorial signaling, cross-pathway inhibition, and the action of a selectively activated scaffold protein. We present a mathematical analysis of simple networks employing each of these mechanisms and show how effective each mechanism is in increasing signal specificity. We also examine the effect of noise in stochastic signaling networks; we prove that the specificity and fidelity of linear and nonlinear networks is not affected by noise. Finally, we talk about the effect of background or basal output levels on specificity and fidelity and show how our definitions can be modified to include high background levels; this section also clarifies how specificity and fidelity can be measured experimentally. In the Discussion, some of the insights derived from our analysis are applied to the yeast mating/invasive growth network.

Figure 2A shows a simple signaling network composed of two signaling pathways, X and Y, initiated by signals x₀(t) and y₀(t), respectively. For pathway X, the time-dependent input signal, x₀(t), activates (that is, causes the production of the active form of) the first component, x₁, which in turn activates the second component, and so on. The level of activation of the final component, x_f, is taken as a measure of pathway output (in Fig. 2 the final component is x₂), which determines the response of the cell to the signal: the cell may move, grow, divide, change its pattern of gene expression or its metabolism, etc.

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Figure 2

A simple network consisting of two parallel cascades with cross talk. The network consists of two pathways, X and Y. Pathway X has three components (x₀, x₁, and x₂); component x₀ is input signal itself, or is activated in manner that is strictly proportional to the input signal. (Because the input signal can have any shape, x₀ can abstractly represent any upstream component, e.g., a receptor, a G-protein, a kinase, etc.) The parameters a₁ and a₂ are activation rate constants, and and are deactivation, or decay, rate constants. For example, a₁ is the rate constant for the activation of x₁ by x₀, and is the rate constant for the deactivation of x₁. (B) Outputs of this network in response to X signaling (that is, X input) and Y signaling. Signal x₀ does not lead to the production of y₂, because none of the components of X can activate or inhibit components of Y. Thus, the specificity of X with respect to Y and fidelity of Y with respect to X are complete (see text for definitions). In contrast, y₁ (which might be a kinase of pathway Y) weakly activates x₂ (which might be a transcription factor for pathway X), at a rate characterized by the “leak constant”, h_leak. Thus, the specificity of Y and the fidelity of X are finite functions of h_leak and other key parameters of the network. In particular, in the text it is shown that network specificity is proportional to 1/h_leak. (C) Depiction of the ratios equal to the specificity of pathway Y and the fidelity of pathway X.

The network shown in Fig. 2 is much simpler than real cellular signaling networks, which may consist of dozens of component proteins. However, a single tier in one of the abstract cascades may be taken to represent several successive steps in a real pathway. For instance, x₀(t) may be taken to represent the plasma concentration of a hormone as a function of time, or the output of a subpathway consisting of hormone-receptor binding, the activation of a receptor-coupled G-protein, and several steps downstream of that. For convenience, we shall frequently call the first component of a pathway the signal, the intermediate component a kinase, and the final component the target.

If the two pathways are not interconnected in any way, then signal x₀ will result in the production of x_f, not y_f; likewise, the activation of pathway Y will neither activate nor inhibit pathway X. Thus, the two pathways, and the network they comprise, will exhibit complete specificity. As discussed above, however, interconnections between pathways often exist; in the network in Fig. 2 there is a connection from y₁ to x₂. In many cases, such interconnections serve a purpose: if it is advantageous for a cell to always have pathway X active whenever pathway Y is active (e.g., because the response evoked by X augments or complements the response evoked by Y), then natural selection may have resulted in a network wired such that Y activates X. On the other hand, cross talk between pathways can be undesirable if it is disadvantageous for pathway Y to influence pathway X (for example, if the response evoked by X is antagonistic or irrelevant to the response evoked by Y).

When pathway Y receives a signal (and X does not), the magnitude of the response of pathway X (if any) provides a measure of the amount of signal crossover. If this crossover represents undesirable leaking, then it should presumably be small compared to both authentic Y signaling (Y output when Y receives a signal) and authentic X signaling (X output when X receives a signal).

Previously we defined two properties, specificity and fidelity, that all pathways in a network must possess to avoid paradoxical situations where the input for a given pathway activates another pathway's output more than its own; or where the output for a given pathway is activated more by another pathway's input than by its own 28.

Let us denote the total output of pathway X when the cell is exposed to an input signal x₀ as (read as “X output given X input”, or simply “X given X”). Similarly, let us define the spurious output of pathway Y when the cell is exposed to signal x₀ as . These quantities should be interpreted as ensemble averages in noisy networks (a detailed analysis is presented further below). The specificity of cascade X (with respect to Y) is the ratio of its authentic output to its spurious output:

(1)

If pathway X is activated by a given signal and this does not result in any output from pathway Y, the specificity of X with respect to Y in response to that signal is infinite, or complete. However, if there is some cross talk between the pathways, then activation of X will result in some output from Y, and the specificity will be finite. If S_X is <1, the input signal for X promotes the output of pathway Y more than its own output.

Similarly, the specificity of cascade Y is defined as follows:

(2)

The overall specificity of the network can be measured by the product

(3)

We say that a pathway or network “has specificity (of degree k)” if for some k>1. Mutual specificity (of degree k) is when both pathways in a network have specificity (of degree k) with respect to each other. The maximum degree of mutual specificity that a network can possibly possess is given by the relationship .

We define the fidelity of X with respect to Y as the total output of X when X receives a signal (and Y does not) divided by the total output of X when Y receives a signal (and X does not). That is, the fidelity of a pathway is its output when given an authentic signal divided by its output in response to a spurious signal.

(4)

A pathway that exhibits fidelity (i.e., F>1) is activated more by its authentic signal than by others. In contrast, if a pathway has fidelity of <1, it is activated more by another pathway's signal than it is by its own. One obvious way for fidelity to be compromised is if a receptor binds promiscuously to several different hormones. However, lack of fidelity may also arise as a consequence of cross talk, as shall be shown further below. As with specificity, the fidelity of the network is the product of the pathway fidelities. Fidelity of degree k and mutual fidelity are also defined similarly to the corresponding expressions for specificity. Note that S_network=F_network, so we will use the term network specificity to mean “network specificity and network fidelity”.

We have found specificity, as defined above, to be a useful analytical concept. However, when considering real biological endpoints, specificity, which compares X_out to Y_out, is essentially comparing apples to oranges. For this reason, fidelity, which compares apples to apples, is perhaps a superior metric when applied to experimental data. More detail concerning how specificity and fidelity can be experimentally measured is presented in the section “Inclusion of background signal levels and experimental measurements” further below.

Alternative definitions of specificity and fidelity that may be useful in some applications would take the form

With this definition, pathway specificities and fidelities would vary in the interval between 0 and 1, inclusive; thus complete fidelity would be characterized by F=1 rather than F equal to infinity, and F below 0.5 would indicate poor fidelity. As another alternative, Schaber et al. 29 defined a measure they called cross talk (C), which is the reciprocal of our fidelity. This metric varies between 0 and infinity, with 0 indicating complete fidelity (no cross talk) and values above 1 indicating poor fidelity. Herein we use the definitions given in Eqs. (1).

As an example of how specificity and fidelity can be calculated in a network of defined architecture, we first consider the simple network shown in Fig. 2. In this network, pathway Y leaks into pathway X, because kinase y₁ is somewhat lacking in substrate selectivity: in addition to phosphorylating its correct target y₂ at a rate proportional to b₂, it also phosphorylates the incorrect target x₂ at rate proportional to h_leak.

Let us denote by the total amount of product x_n when the cell is exposed to signal x₀ but not to signal y₀. Similarly, denotes the total amount of y_n under the action of signal x₀. Let x_f and y_f denote the final products of pathways X and Y, respectively, so that is another way of writing . For the purposes of the following exposition, we presume that the production of x_f in response to signal y₀ is undesirable.

Our approach, similar to that of Heinrich et al. 30, is to model the enzymatic reactions of signaling pathways using equations that are simplifications of the standard mass action or Michaelis-Menten formulations. These simplifications are made so that exact analytical solutions of the equations can be obtained in most cases. In particular, we assume that the pathways are weakly activated, meaning that the level of component activation is low compared to the total amount of that component in the cell. (In the Supplementary Material , we demonstrate that some of the key results hold even when pathways are strongly activated.) The assumption of weak activation allows signaling cascades to be modeled as a linear system 30,31. For instance, when pathway X is on (and Y is off), the dynamics of signaling in pathway X can be expressed as a simple linear system of ordinary differential equations (ODEs):

(5)

(6)

Here, x₀(t) is the signal function, and x₁ and x₂ are concentrations of the active species of these components at a given moment of time. The parameters a₁ and a₂ are activation rate constants; a₂ is proportional to the rate at which kinase x₁ activates (phosphorylates) target x₂. Similarly, and are deactivation (or decay) rate constants, and can be thought of as representing phosphatase activity or protein degradation, for example. The term is a shorthand notation for , the rate of change of component x₁ at a particular moment in time. This is equal to the amount of x₁ being created minus the amount being destroyed at that time. The former is equal to the magnitude of signal x₀ multiplied by the rate constant a₁; the latter is equal to the concentration of x₁ multiplied by the decay rate constant . Equations (5) can be interpreted as equations for average values of the variables, and can be rigorously derived from a stochastic process; this is done later in the article.

The solution of Eqs. (5) is obtained by integrating both sides from zero to infinity; resulting in the left-hand side of each equation being replaced by zero and x₁ and x₂ being replaced by and , respectively. Rearrangement then yields

(7)

Let us hereafter refer to the level of signal flux to the intermediate component (i.e., x₁ or y₁ in the examples herein) as the signal strength; this will be a product of the magnitude of the input signal and the rate coefficient(s) for the upstream step(s), or in the above example.

The dynamics of pathway Y signaling under the action of y₀ can be similarly expressed as

(8)

(9)

From these it can be determined that

(10)

Furthermore, it is obvious by inspection that . Thus, the expressions for S_X and F_Y have zero in the denominator, so X can be considered to have infinite, or complete, specificity (with respect to Y), and Y can be considered to have complete fidelity. The calculation of requires modifying Eq. (6) so that it becomes (Eq. (6)):

Note that the first term of Eq. (6) is zero when pathway X is off and the second term is zero when Y is off. From Eqs. 6a and 8 it can be determined that

(11)

(12)

Hence, both quantities are decreasing functions of the “leakage rate” h_leak, and will be very large if h_leak is very small.

Network specificity is undefined when one pathway has complete specificity; only if there is bidirectional crossover does it make sense to calculate network specificity. To add crossover in the other direction, assume that kinase x₁ also lacks complete selectivity, and phosphorylates substrate y2 at rate j_leak. This results in the architecture shown in Fig. 3. In this case, Eq. (12) still holds for S_Y and F_X, and in addition

(13)

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Figure 3

The output of four signaling networks exhibiting different degrees of specificity and fidelity. The bars on the graph represent the total output of signaling pathways X and Y (X_out and Y_out, respectively) under the condition where pathway X is receiving a signal and pathway Y is not (X_in), and visa versa (Y_in). The results correspond to solutions of the network described in Eqs. (5). All parameter values were set equal to 1 except as specified below. The thickness of the links connecting the component nodes is proportional to the rate coefficient for that reaction (see Figure 2A; j is the coefficient for the connection from x₁ to y₂). (A) This network exhibits both mutual specificity and mutual fidelity. Networks such as this are presumably the most useful to the organism. Rate coefficients are a₁=2, a₂=3, b₁=2, b₂=2.5, h=0.5, j=1. (B) This network displays mutual specificity but not mutual fidelity. Pathway Y does not exhibit fidelity with respect to pathway X, because Y output when Y is on is less than Y output when X is on. Parameter values are a₁=2, a₂=3, b₁=1, b₂=1.5, h=0.5, j=1. (C) This network possesses mutual fidelity, but pathway Y does not exhibit specificity with respect to X. Parameter values are a₁=2, a₂=3, b₁=2, b₂=1.5, h=2, j=1. (D) This network possesses neither mutual specificity nor mutual fidelity. Parameter values are a₁=2, a₂=1, b₁=2, b₂=1.5, h=2, j=3.

The ratios a₂/j_leak and b₂/h_leak can be taken as measures of the selectivity of kinases x₁ and y₁, respectively, for their correct versus incorrect targets. As can be seen, the specificity of the network in this case is equal to the product of these selectivities. Moreover, both mutual specificity and mutual fidelity are rather easily achieved. For instance, in the case of symmetric network parameters (, etc.), the conditions for both are a₂/j_leak>1, b₂/h_leak>1. (To be precise, a₂/j_leak≥k; b₂/h_leak≥k; k>1.)

Fig. 3 shows some example solutions of this network under the action of signal x₀ (“X_in”) and signal y₀ (“Y_in”), and the resulting specificity and fidelity values. The network in Figure 3A possesses both mutual specificity and mutual fidelity. As can be seen, however, one can envisage a network with mutual specificity but without mutual fidelity (Figure 3B), and visa versa (Figure 3C). Furthermore, a given pathway can exhibit both specificity and fidelity, or only one or the other, or neither (Figure 3D).

For the remainder of this article, we will examine the situation where two signaling pathways share one or more common elements (see Figure 4A). Without any further assumptions, this class of networks can be represented by the simple architecture shown in Figure 4A, and it can be expressed as the following ODEs:

(14)

(15)

(16)

We refer to this as the “basic architecture”, because the network lacks any enhancements designed to promote specificity. In the cases we will consider, we assume the network only receives one of the two signals at a time. Thus, if x₀(t) is positive for some duration of time, then y₀(t) is identically zero, and visa versa; either the first or second term of Eq. (14) is equal to zero, depending upon which of the two signals the network is receiving. Assume that signals x₀(t) and y₀(t) have the duration t_x and t_y, respectively, and average magnitudes equal to the constants and . Then, when X is “on” and Y is “off”,

(17)

Furthermore, when pathway Y is “on” and X is “off” (that is, y₀(t)>0 and x₀(t)=0),

(18)

From these expressions we can calculate that

(19)

These quantities are quite easy to understand intuitively. The expressions for fidelity are simply ratios of signal strength multiplied by signal duration, that is, ratios of the total amount of signal flowing into the shared component. In contrast, the quantities for specificity report on signaling downstream of the activation of the shared component. In the case of S_X, the numerator contains a coefficient (a₂) that positively influences X_out and a coefficient () that negatively influences Y_out, whereas the coefficients in the denominator act conversely.

Note from Eq. (19) that S_X is the reciprocal of S_Y, and F_X is the reciprocal of F_Y. Thus, because S_network=S_XS_Y=F_XF_Y, S_network=1. Thus, the basic architecture does not exhibit overall network specificity, nor does it exhibit mutual specificity or mutual fidelity 28.

Real cellular signaling networks that share components contain one or more insulating mechanisms that are thought to contribute to specificity and fidelity, some of which are shown in Fig. 4. First, in combinatorial signaling, the simultaneous action of two or more different signals may be required to evoke a response, so that the output of a pathway is determined by the combination of signals acting on a network (Figure 4B). For example, the survival of epithelial cells requires two signals, one provided by growth factors and transmitted by the MAPK pathway, and one provided by cell attachment 32, and Wnt and BMP signals combine to determine whether neural crest stem cells will differentiate 33. Another type of combinatorial signaling occurs when a pathway branches into two subpathways (one that contains shared components and one that doesn’t) that are reintegrated at a point further downstream, so that the response to a given signal is determined by the combination of subpathways activated by that signal (Figure 4C) 34. Combinatorial signaling requires that a downstream component (such as x₂ in Figure 4BC) is able to act as a molecular “AND gate” or “coincidence detector” that integrates two separate inputs 35,36. One of these inputs may be a component that is shared with another pathway (such as kinase x₁, which is shared with pathway Y). If the other input is not shared with Y, then this may be exploited to enhance the specificity of the XY network. Some examples of proteins that function as signal integrators include Smad1, which integrates MAPK and TGFβ signals 37, and the estrogen receptor, which integrates MAPK and estrogen hormone signals 38. In addition, DNA regulatory elements such as the Drosophila eve-skipped enhancer can also act as signal integrators 39.

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Figure 4

Signaling network with shared components. (A) The simplest such network, herein referred to as the “basic architecture”, shown with activation and decay rate constants. Component x₁ is common to pathways X and Y. This network cannot achieve specificity and fidelity. (B–F) Elaborations to the basic architecture that are found in nature and have been proposed to promote specificity. (B) Combinatorial signaling with an independent, parallel input provided by pathway Z. (C) Combinatorial signaling via the branching and reintegration of pathway X. (D) Cross-pathway inhibition. (E) Compartmentalization. (F) The action of a scaffold protein.

Indeed, combinatorial signaling is used in the yeast mating and invasive growth signaling network to regulate a subset of filamentation genes during invasive growth. The first signal is relayed via the cell surface protein Msb2, which senses localized turgor pressure (or some other mechanical force, presumably) and sends a signal via the MAPK cascade to activate the Tec1 transcription factor 40. A second, glucose limitation signal is sensed by a the Snf1 protein kinase, which inhibits a transcriptional repressor known as Nrg 41. Both Tec1 and Nrg bind to the promoters of certain genes required for invasive growth, and efficient activation of these genes requires both Tec1 activation and removal of Nrg-dependent repression 15. Thus, optimal haploid invasive growth requires both a mechanical signal and a glucose limitation signal.

To add combinatorial signaling for pathway X, we modify Eq. (15) of the basic architecture characterized above (Eqs. (14)) by adding a single term R[x₀], so that (Eq. (15)):

Here, R[x₀] represents the combinatorial input. Thus, target x₂ is the signal integrator, or “AND gate”: x₂ activity depends on two separate inputs, R and x₁. If either input is zero then x₂ is also zero. Note the case when the coefficient is identical to the basic architecture. To add the influence of an independent, parallel pathway Z, as in Figure 4B, we set

(20)

Here we assume that Z is activated concurrently with X, because the cell is usually exposed to both signals at the same time. Further, we assume that Z is mostly, but not completely, off when X is off. The basal activity of Z when X is off is characterized by the leakage rate k_leak, and provides a “back door” via which Y can leak into X.

Alternatively, if the situation shown in Figure 4C applies, where X branches and is reintegrated at x₂, then the flux through the X-dedicated subpathway will be proportional to the signal x₀. This can be represented by setting R[x₀]=x₀(t)+k_leak. Here the leakage constant k_leak represents the basal activity of the X-dedicated subpathway, and the requirement that can be dropped. Because the activation of the shared kinase x₁ is also proportional to signal x₀, branching and reintegration leads to “signal multiplication” with itself, a phenomenon that has been examined for a single pathway by Heinrich et al. 30.

Suppose for simplicity that signal x₀(t) is a square pulse of amplitude and length t_x, and y₀(t) is a square pulse of amplitude and length t_y. Specificity for all three models (Figure 4AC) are then given by

(21)

Here is a positive quantity that becomes insignificant if the duration of the signal x₀ exceeds 1/d₁, the characteristic time for the deactivation of kinase x₁. We can see that for shorter signals, specificity is always lower, and it reaches a saturation level for relatively long x₀; this level is given by

(22)

In both types of combinatorial signaling, specificity is inversely proportional to the amount of leakage, and by making k_leak small, it is possible to obtain arbitrarily high levels of network specificity.

For all three models, the expressions for fidelity are:

(23)

In summary, compared to the basic architecture, combinatorial signaling raises F_X, S_Y, and S_network, and has no effect on F_Y. Under the branching/reintegration scheme, S_X is also increased.

Interestingly, with combinatorial signaling there are obstacles to obtaining mutual specificity and fidelity that place additional requirements on the characteristics of the network. In the case of an independent parallel input, achieving mutual specificity of degree k requires both k_leak≤1/k² and , where . Clearly, the second condition is impossible in the case of symmetric network parameters. Note that is the “local sensitivity coefficient” for x₂ production—the percent of change in x₂ caused by a 1% change in x₁ at steady state 42. Similarly, is the local sensitivity coefficient for y₂ generation; thus α measures relative signal transfer from x₁ to x₂ vs. y₂. To obtain mutual specificity of a reasonably high degree, this must be correspondingly high. One way to achieve this is to reduce b₂, the rate constant for the phosphorylation of target y₂ by kinase x₁, or in other words, to make a y₂ a poor substrate for x₁. This analysis suggests that maximizing the performance objective of mutual specificity may favor a seemingly paradoxical situation where an authentic substrate of a kinase is a poor target for that kinase. Mutual fidelity also cannot be obtained with symmetric network parameters and signal inputs. Achieving mutual fidelity requires greater strength or/and duration of the Y signal compared to the X signal, and low value of k_leak. The branching/reintegration scheme (Figure 4C) makes it easier to achieve mutual specificity and mutual fidelity, but the latter still requires a stronger or longer Y signal feeding into kinase x₁.

Fig. 5 shows a typical design of an optimized network featuring combinatorial signaling. The figure illustrates how achieving the goal of specificity shapes the network design so that the rate coefficients leading from y₀ to x₁ and from x₁ to x₂ are large, whereas those from x₀ to x₁ and from x₁ to y₂ are small.

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Figure 5

Representative network featuring the “combinatorial signaling” insulating mechanism with an independent, parallel input, as shown in Figure 4B. The abstract action of the molecular AND gate is represented by the symbol inscribed with an “A”. The thickness of the lines connecting the components represents the magnitudes of the rate coefficients. Network parameters are k_leak=0.1, x₀=1, y₀ =1, t_X=1, t_y =1. Rate coefficients are . Deactivation rates are not shown and were set equal to 1. The thickness of the arrows leading from one component to another represent the signal flux through that point of the network under the action of signal x₀ (A) or y₀ (B). The design shown optimizes network specificity, mutual specificity, and mutual fidelity, given parameters a₁, b₁, a₂, b₂ chosen from the range {0.5–4}.

The obstacles to obtaining mutual specificity and fidelity in the above model arise because an insulating mechanism has been added to just one of the two pathways in the network. If a combinatorial input was also provided to pathway Y, then it would be straightforward to achieve both mutual specificity and fidelity, even in the case of symmetric network parameters, provided only that the leakage constants were small.

Cross-pathway inhibition (Figure 4D) occurs when a downstream component of pathway X inhibits a downstream component of pathway Y. An example of this type of inhibition has recently been discovered in yeast, where it was shown that Fus3, the MAP kinase in the mating pathway, phosphorylates Tec1, a transcriptional regulator for invasive growth, and thereby accelerates the degradation of Tec1 21,22,43,44. This promotes specificity during mating by preventing Kss1^MAPK, which is also activated by mating pheromone, from activating Tec1-dependent transcription of filamentation genes (Fig. 6). This situation can be represented by the general architecture shown in Figure 4D, where component x₂ inhibits y₂. To model inhibition, we assume that the effective decay rate of the inhibited component is a growing function of the concentration of the inhibiting component. This results in a modification of Eq. (16) (Eq. (16)):

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Figure 6

The “cross-pathway inhibition” insulating mechanism. (A) The simplest abstract network featuring cross-pathway inhibition. (B) A real network containing cross-pathway inhibition, the yeast mating/invasive growth signaling network.

Here, g is a rate coefficient that relates the amount of x₂ to the increased decay of y₂. Clearly, the mechanism of inhibition is nonlinear by nature. However, it is possible to solve the system of equations (Eqs. (14)) under the assumption that the signals x₀ and y₀ are square pulses of amplitudes and and lengths t_x and t_y, which are long compared to the half-lives of the activated proteins, (i.e., the steady-state approach). This approach yields

(24)

Thus, compared to the basic architecture (see Eq. (19)), cross-pathway inhibition increases S_X and decreases S_Y. The specificity of the network will be >1 only if ; that is, the signal strength for pathway X must be stronger than that for pathway Y.

The condition for mutual specificity of degree k is equivalent to two simultaneous inequalities,

The fidelity values of the two cascades are given by

(25)

Thus, F_X is the same as in the basic architecture and F_Y will be greater than in the basic architecture only if . Mutual fidelity is even more difficult to achieve than mutual specificity. However, it is possible to achieve mutual fidelity to some degree by increasing the strength of signal x₀ compared to the strength of signal y₀ while keeping signal y₀ sufficiently long compared to signal x_0. In the “best” case, where the decay rates are very small compared to the other constants, mutual fidelity of degree k could be obtained providing and . Hence, the requirement of mutual fidelity imposes conditions on both the relative strength and duration of the input signals. Mutual fidelity is impossible for signals of equal length or equal strength.

Fig. 7 depicts a representative network featuring cross-pathway inhibition of Y by X as its only insulating mechanisms. The network achieves some degree of mutual specificity but cannot achieve mutual fidelity, because the signal durations are similar. Obtaining mutual specificity constrains the network design so that the rate coefficients leading from x₀ to x₁ and from x₁ to y₂ are large, whereas those from y₀ to x₁ and from x₁ to x₂ are small.

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Figure 7

Representative network featuring cross-pathway inhibition as its only insulating mechanism. The thickness of the lines connecting the components represents the magnitudes of the rate coefficients. Signal magnitudes and durations are x₀=1, y₀ =1, t_X=1, t_y =1. Rate coefficients are a₁=4, a₂=1, b₁=0.5, b₂=4, g=2. Deactivation rates are not shown and were set equal to 1. The thickness of the arrows leading from one component to another represent the signal flux through that point of the network under the action of signal x₀ (A) or y₀ (B). The design shown optimizes mutual specificity given parameters a₁, b₁, a₂, b₂, g chosen from the set {0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4}.

Elsewhere we analyzed the insulating mechanism of compartmentalization and compared it to the sequestering function of scaffold proteins 28. In compartmentalization, different pathways are localized to different compartments or regions of the cell, such as the nucleus and the cytosol. Leaking between pathways can only occur if a shared component, while active, moves from one compartment to another (Figure 4E).

Signaling scaffolds are proteins that bind to two or more consecutively acting components of a pathway and accelerate the rate of reaction between them (Figure 4F). For example, yeast Ste5 protein is a scaffold protein in the mating pathway. It binds to the G-protein that is an upstream component of the mating pathway, to the shared intermediate kinases Ste11^MAP3K and Ste7^MAP2K, and to the mating-specific downstream kinase Fus3^MAPK45. As such, it enables a pathway-specific upstream component to activate the shared kinases, and then helps the shared kinases to activate a pathway-specific downstream kinase (Figure 8A). This type of mechanism may enhance specificity if the movement of the active shared kinases on and off the scaffold is limited. In particular, if the reactions between the kinases can only happen while bound to the scaffold (or are much more efficient on scaffold than off scaffold), then this sequestering action of the scaffold is formally equivalent to compartmentalization, and both mechanisms can be represented by the same set of equations, as follows:

(26)

(27)

(28)

(29)

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Figure 8

Action of a selectively activated scaffold. The cartoon is based upon a model recently proposed for the action of yeast Ste5 23. (A) During mating, Ste5 is activated by the mating-pathway-specific G-protein. Activated Ste5 is competent to act as a scaffold by binding to the shared kinases Ste11^MAP3K and Ste7^MAP2K and promoting the Ste7-dependent activation of the mating-pathway-specific kinase Fus3^MAPK. (B) When cells are not mating, Ste5 is in an inactive conformation and cannot channel signals from the shared kinases to Fus3. Thus, even if the cell is undergoing invasive growth or experiencing other stimuli that activate some of the shared kinases, the mating response is not activated. In these situations, however, it is likely that some minor fraction of the Ste5 in the cell is in the active conformation; in the mathematical model this fraction is represented by the parameter k_leak.

Here, is the concentration of active kinase x₁ in the Nucleus (or aNchored, or oN, the scaffold) and is the concentration of active kinase x₁ in the Cytosol; D_out is the rate constant at which x₁ exits the nucleus/scaffold and enters the cytosol, and D_in is the rate at which x₁ in the cytosol enters the nucleus or binds to the scaffold. The parameters and are the deactivation constants for x₁ in the nucleus/on the scaffold and in the cytosol, respectively; for example, if x₁ is activated by phosphorylation, can denote the rate of dephosphorylation of scaffold-bound x₁. Note from Eq. (29) that y₂ only arises via cytosolic x₁, and note from Eq. (27) that, when X is on and Y is off, cytosolic x₁ only arises by the relocation/dissociation of nuclear/scaffold-activated x₁.

The equations for specificity and fidelity are:

(30)

(31)

(32)

It can be seen that network specificity is >1, and thus is greater than in the basic architecture. Each of four quantities, S_X, S_Y, F_X, and F_Y, differ from the corresponding values in the basic architecture by a factor that is a ratio of an exchange rate constant D plus a decay constant d divided by an exchange rate constant. For example, S_X differs by a factor of from the basic architecture result. This makes sense: S_X is favored by having small D_out (so that very little x₁ leaks out of the nucleus/off the scaffold), and large D_in and (so that any x₁ that does leak out is likely to either move back in or decay). In the case of symmetric network parameters, the condition guarantees mutual specificity and fidelity of the cascades X and Y. If D_in=D_out=D and , then this condition is automatically satisfied, and provides mutual specificity and fidelity of degree k=(d₁+D)/D, where d₁ is the lesser of . Compartmentalization or spatial separation makes “leakage” ineffective as long as the rates of leakage balance each other, and are small compared to the deactivation rates.

Another way in which a scaffold might promote specificity is if the scaffold is only in a conformation capable of productively binding the shared kinase(s) during authentic signaling. This mechanism has been termed “selective activation” 23. For example, it has been proposed that the yeast Ste5 scaffold is found in an active conformation only during mating (Figure 8A). During invasive growth signaling, Ste5 exists in a “closed” conformation (or is localized in the wrong place or is for some other reason inactive) and so cannot channel signals from the shared kinase Ste7^MAP2K to the mating-specific kinase Fus3^MAPK (Figure 8B) 23,46.

To model the selective activation of a scaffold, we modify Eqs. (26) so that they become (Eqs. 26a and 27a):

(33)

(34)

Both F_X and F_Y also increase (provided k_leak<1), and only S_X is unchanged by the addition of selective activation. Note that Eqs. (33) are strictly valid only in the steady state, i.e., if the signal durations are long compared to the inverse of the eigenvalues of the linear matrix that appears in system (Eqs. 26a and 27a).

In the Supplementary Material , a more elaborate model of scaffolding is presented that is described by a nonlinear system of six differential equations, and yet is still solvable. This model includes the formation of a complex between inactive kinase x₁ and the scaffold and the activation of x₁ on the scaffold, allows both events to be dependent on the signal, and differs in several other details from the above models. Despite these differences, the expressions for network and pathway specificity and fidelity are very similar to those given above. Thus the simpler models appear to capture some of the key features of the specificity-promoting qualities of scaffold proteins.

In summary, scaffold proteins can enhance specificity both by sequestration, a mechanism that resembles compartmentalization, and by selective activation, a mechanism that resembles combinatorial signaling. As such, a scaffold can in principal provide a highly effective insulating mechanism. Indeed, scaffolds are often spatially localized, which would result in an insulating mechanism more effective than either compartmentalization or scaffolding per se. To what extent scaffold proteins use sequestration and/or selective activation to enhance specificity is an area of active experimental investigation. A recent study of the yeast Ste5 protein suggests that selective activation may be more important than sequestration for this scaffold 23. In terms of the model, Ste5 might be considered to be a scaffold for which D_in and D_out are high relative to and , and k_leak is low.

Signaling networks are noisy systems, and there has been much work on modeling noise in chemical signaling (see, for example, 47,48,49,50,51,52). In this section we examine the effects of noise on specificity and fidelity. We conclude that the effects of noise usually “average out” when specificity and fidelity are calculated, and therefore that the simpler deterministic treatment presented above is generally valid.

There are two sources of noise: internal and external 53. Internal noise is due to the stochastic nature of the collisions and reactions of the proteins involved. External noise includes all sources of noise not directly related to the proteins involved in our description, such as noise from other molecules that affect the system but that are not explicitly described in the model. To model the effect of noise on specificity of signaling networks, we will first derive a chemical Langevin equation 54 to account for the internal noise, and describe how external noise can also be included.

Let us start from a simple linear cascade, , where species x₁ and x₂ have the decay rates of and , respectively. Let us denote by i, j, and k the number of molecules of proteins x₀, x₁ and x₂, respectively. We assume the following Poisson process: in an infinitesimal time interval, Δt, the following changes can occur:

Let us denote by the probability to have j molecules of type x₁ and k molecules of type x₂ at time t. We have the following Kolmogorov (master) equation:

(35)

We can define the average amount of each of the species x₁ and x₂ as

Equations for these quantities can be derived from Eq. (35); they are

These are identical to Eqs. (5). Following Gillespie's argument 54, we can write down a continuous (diffusion) approximation of the master equation. Defining continuous variables X₁ and X₂ such that , and expanding the terms in the right-hand side of the master equation into a Taylor series up to the second order, we obtain the following Fokker-Planck equation:

(36)

Here, A_1,A₂, B₁, and B₂ are the drift and diffusion coefficients, respectively. Equation (36), just like the master equation, describes the evolution of the probability distribution function, . For individual stochastic trajectories, a different description has to be developed. Using the expressions for the drift and diffusion coefficients, we can derive the chemical Langevin equation for the stochastic variables:

(37)

There are many ways to incorporate external noise in the system. However, to illustrate the effect of external noise, it suffices to use a simple description. Here we assume that the noise affects the variable x₀, the input signal, and that it propagates down the cascade by means of Eq. (37). The input signal can be thought of as the solution of the Ornstein-Uhlenbeck-type equation,

This derivation can be easily generalized to more complicated or even nonlinear networks. The nonlinearities in the activation-deactivation coefficients are simply carried over to the drift and diffusion terms. To give an example, we present the SDEs for two parallel cascades with a shared element and a nonlinearity:

Solutions of these equations have the form

Now we can introduce definitions of specificity and fidelity. Using a similar approach to that taken in the deterministic case, we define as the total amount of final product X_f when the cell is exposed to signal x₀ but not to signal y₀. The specificity and fidelity of channel X are given by

(38)

To calculate the specificity of the X channel for the stochastic system, we first integrate the SDEs to obtain

(39)

Similar expressions can be derived for the average amount of all species. In fact, these expressions are not different from the deterministic ones obtained previously. As a result the stochastic effects do not change the specificity and fidelity calculated using the deterministic approach. This is not surprising, because this study is concerned with global characteristics of the system, and the noise usually does not influence ensemble-averaged quantities.

Before we go on, we would like to comment on the averaging procedure used in the above definition. The specificity and fidelity are evaluated by first averaging both signals (under the condition that one input is on and the other is off), and then a ratio is formed. An alternative way would be to evaluate the following: