Different cellular signal transduction pathways are often interconnected, so that the potential for undesirable crosstalk between pathways exists. Nevertheless, signaling networks have evolved that maintain specificity from signal to cellular response. Here, we develop a framework for the analysis of networks containing two or more interconnected signaling pathways. We define two properties, specificity and fidelity, that all pathways in a network must possess in order to avoid paradoxical situations where one pathway activates another pathway's output, or responds to another pathway's input, more than its own. In unembellished networks that share components, it is impossible for all pathways to have both mutual specificity and mutual fidelity. However, inclusion of either of two related insulating mechanisms—compartmentalization or the action of a scaffold protein—allows both properties to be achieved, provided deactivation rates are fast compared to exchange rates.
Cells sense and respond to a wide variety of chemical messages, such as hormones like insulin and adrenalin, which originate from other cells or from the environment. Yet, cells use only a limited number of intracellular signaling proteins to transduce this multitude of signals. For instance, some key intermediate modules, such as the mitogen‐activated protein kinase (MAPK) cascade, are activated by an astonishingly high percentage of known stimuli (Lewis et al, 1998). Hence, different signals are often transmitted by common components, yet elicit distinct (and appropriate) outcomes. An important unsolved problem in cell biology is to understand how specificity from signal to cellular response is maintained between different signal transduction pathways that share similar (or identical) components, particularly when this occurs in the same cell (Schaeffer and Weber, 1999; Tan and Kim, 1999; Pawson, 2004).
Results and discussion
Simple signaling networks
Figure 1A shows a schematic of a simple signaling network, composed of two pathways, X and Y. Each pathway in the network has a unique input and a unique output. The input for pathway X is designated x0, and can be taken to represent both the signal itself and its receptor; the input for pathway Y is designated y0. We assume that the network is exposed to only one signal at a time. The activated species of the downstream elements of pathway X are designated x1 and x2, those of pathway Y are y1 and y2. When a signal x0 is present, x0 ‘activates’ (i.e. causes the production of) component x1 (which might be a protein kinase or a kinase cascade); x1 in turn activates x2 (which might be a terminal kinase or a target transcription factor). Pathway components are also deactivated: proteins that are activated by being phosphorylated by a protein kinase are deactivated when that phosphate group is removed by a protein phosphatase, for instance. Figure 1B shows the typical mound‐shaped curve of the time course of activation of a given component in response to a square‐pulse input signal. The area under such a curve for the final component of a given pathway can be taken as a measure of that pathway's total output. Let us denote the total output of pathway X when the cell is exposed to an input signal x0 as Xout∣Xin (read as ‘X output given X input’, or simply ‘X given X’).
Interconnections between pathways often serve a useful purpose (Schwartz and Baron, 1999), but here we concern ourselves with undesirable crosstalk, or ‘leaking’. In the Figure 1 network, pathway Y leaks into pathway X, because kinase y1 is somewhat lacking in substrate selectivity: in addition to phosphorylating its correct target y2, it also phosphorylates the incorrect target x2. Hence, when the network is stimulated by signal y0, in addition to the authentic output Yout∣Yin there is some spurious output Xout∣Yin.
Definitions of specificity and fidelity
We define the specificity of cascade X as the ratio of its authentic output to its spurious output:
Thus (as in Figure 1), if pathway X is activated by a given signal and this does not affect the output from pathway Y, the specificity of X with respect to Y in response to that signal is infinite, or complete.
Similarly, the specificity of cascade Y is of the form
In Figure 1, since the action of signal y0 will result in some output from X, the specificity of Y with respect to X is finite (see Figure 1C). Indeed, if SY were less than 1, it would mean that the signal for Y was actually promoting the output of pathway X more than its own output.
The fidelity of a pathway is its output when given an authentic signal divided by its output in response to a spurious signal (see Figure 1C):
Thus, 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 less than 1, it is activated more by another pathways’ signal than it is by its own. In the Figure 1 network, FY is complete, while FX is finite.
Cascades that share components
In many cases, two signaling pathways share one or more common elements (see Figure 2A). One well‐known example is in mammalian PC12 cells, where treatment with epidermal growth factor (EFG) causes the cells to proliferate, whereas treatment with nerve growth factor (NFG) causes the cells to differentiate and sprout neurites, yet both growth factors signal through the same MAPK cascade (Marshall, 1995). Another example is in baker's and brewer's yeast (Saccharomyces cerevisiae), where three distinct signaling pathways (mating, invasive growth and osmotic stress response) share elements of the same MAP kinase cascade (van Drogen and Peter, 2002). Experimental data indicate that pathways can be well insulated from one another despite sharing components: treatment of PC12 cells with EGF does not cause them to sprout neurites, and stimulation of yeast with mating pheromone does not activate the stress response, for example (Schaeffer and Weber, 1999; van Drogen and Peter, 2002; Vaudry et al, 2002).
This class of networks can be represented by the ‘basic architecture’ shown in Figure 2A. Here, the parameters a1 and a2 are activation rate constants for pathway X; a2 is the rate at which kinase x1 activates (phosphorylates) target x2. Similarly, b1 and b2 are activation rate constants for pathway Y. Finally, d1x, d2x and d2y are deactivation rate constants, and can be thought of as representing phosphatase activity, for example. Assuming that both pathways in the network are weakly activated (Heinrich et al, 2002; Chaves et al, 2004), the network can be modeled by a simple system of three linear ordinary differential equations (ODEs) that can be solved to yield precise analytical expressions for pathway outputs, specificities and fidelities in terms of the network parameters (see Table I; also, see Supplementary information for detailed solutions). It can be seen (see Table I) that the specificities of the pathways are independent of the signal strength, and indeed of all parameters that lie upstream or at the level of the shared component. Pathway fidelities, in contrast, depend strongly upon the relative signal strengths and upon the values of upstream parameters. Hence, the two performance metrics, specificity and fidelity, depend on different characteristics of network design. Indeed, it is easy to choose parameters that provide one pathway with specificity but not fidelity.
We define network specificity as the product of the pathway specificities:
(Note that network fidelity, the product of the pathway fidelities, is always exactly equal to network specificity.) Snetwork provides an indication of the specificity intrinsic in the network architecture. Intuitively, it would seem that the basic architecture does not possess intrinsic specificity. Indeed, it can be seen from Table I that, for the basic architecture, SX is the reciprocal of SY, and FX is the reciprocal of FY, so that Snetwork=Fnetwork=1. The specificity of pathway X can be increased by changing the magnitude of certain parameters (increasing a2 or decreasing b2, for example), but in so doing the specificity of Y decreases correspondingly (see Figure 2B and C).
Two other useful network measurements are mutual specificity (and mutual fidelity), properties that exist if all pathways in the network have specificity (fidelity) greater than 1. The basic architecture never exhibits mutual specificity or mutual fidelity.
Analysis of insulating mechanisms: compartmentalization
Real cellular signaling networks that share components typically contain one or more insulating mechanisms that are thought to contribute to specificity and fidelity (Tan and Kim, 1999; Schwartz and Madhani, 2004). In compartmentalization, different pathways are localized to different cellular compartments, or to different spatial locations within the cell (Figure 2D) (Smith and Scott, 2002; White and Anderson, 2005). The extent of leaking between the two pathways is determined by the efficiency of compartmentalization. For example, assume that the pathway‐specific components of pathway X are localized to the nucleus, while those of pathway Y are localized to the cytosol. Although the shared kinase, x1, is found in both compartments (x1N is the nuclear pool and x1C is the cytosolic pool), x1 activated by x0 in the nucleus is likely to encounter target x2, which is also in the nucleus; it will only encounter target y2 if it diffuses into the cytosol before it is deactivated. Thus, crossover between the two pathways happens when kinase x1 leaks in or out of the nucleus. Dx is the coefficient for the rate at which x1 exits the nucleus and enters the cytosol, and Dy is the rate constant for x1 leaving the cytosol and entering the nucleus. Dx and Dy can be considered as pseudo‐diffusion rate constants, or exchange rate constants. The parameters d1x and d1y are the deactivation constants for x1 in the nucleus and cytosol, respectively.
Again, assuming weak activation, the network can be modeled with linear ODEs and precise solutions for specificity and fidelity obtained (see Table I and Supplementary information). The specificity of this network is
It can be seen that network specificity is greater than in the basic architecture, and is maximized if the exchange rates Dx and Dy are small compared to the deactivation rates d1x and d1y. Compartmentalization can also provide both mutual specificity and mutual fidelity, as long as the exchange rates balance each other (see Table I). The limiting case where Dx=Dy=0 is equivalent to two noninteracting cascades with complete specificity and fidelity. If the leakage coefficients become very large (Dx, Dy → ∞), we again have a fully connected system with a shared element, equivalent to the basic architecture, and Snetwork=Fnetwork=1.
Role of scaffold proteins
Scaffold proteins, defined here as proteins that bind to two or more consecutively acting components in a pathway, have been shown to enhance the efficiency of signaling, and have also been proposed to augment specificity by several mechanisms (Whitmarsh and Davis, 1998; Levchenko et al, 2000; Burack et al, 2002; Flatauer et al, 2005). In particular, by binding to multiple components of a given pathway, scaffolds may create the equivalent of ‘micro‐compartments’ (Harris et al, 2001). That is, if the reactions between those components can only happen on the scaffold (or are much more efficient on the scaffold), then it is as if these scaffolded reactions occur in their own compartment, sequestered away from reactions occurring off‐scaffold. In this way, scaffolds may prevent their bound components that are shared with other pathways from straying into those pathways, and protect them from intrusions from those pathways.
To model this sequestration mechanism, we use the equations for compartmentalization, with the meaning of some of the terms interpreted differently (see Figure 2E and Table I). First, x1N (aNchored x1) is interpreted to represent kinase x1 bound to the scaffold and x1C (Cytosolic x1) is unbound x1, free in solution in the cytosol. The equation for dx1N/dt then indicates that the activation of kinase x1 by signal x0 occurs on the scaffold and not in solution, while the equation for dx2/dt indicates that the activation of target x2 by kinase x1 also occurs only on the scaffold. In contrast, the corresponding reactions for pathway Y can occur only in solution and not on the scaffold. Dx is the rate constant for the dissociation of x1 from the scaffold, and Dy is a first‐order association constant for the binding of cytosolic x1 to the scaffolded complex. Leaking between pathways X and Y can occur if x0‐activated x1 dissociates from the scaffold and encounters y2, or if x1 that was activated by y0 in the cytosol binds to the scaffold (see Figure 2E).
The previous results (see equation (5) and Table I) for specificity and fidelity under compartmentalization also apply to scaffolding: SX is promoted by a low rate of dissociation of kinase x1 from the scaffold and a high rate of rebinding; however, these factors reduce SY. Obtaining network specificity again requires that deactivation rates be fast relative to exchange rates, so that, for instance, any x1 that dissociates from the scaffold will be deactivated before it encounters y2. In this model, d1x represents the deactivation of kinase x1 on the scaffold. Interestingly, one way in which it has been proposed that scaffold proteins might enhance signal transmission is by protecting their bound kinases from the action of phosphatases (Levchenko et al, 2000; Heinrich et al, 2002), equivalent to lowering d1x to close to or equal to zero. Although this might indeed enhance the speed, duration and amplitude of X signaling (Heinrich et al, 2002), it would lower both SY, FY and network specificity.
Here, we presented a framework for the analysis of interconnected biochemical pathways. We defined the specificity of a pathway as the ratio of its authentic output to its spurious output, and the fidelity of a pathway as its output when given an authentic signal divided by its output in response to a spurious signal. These definitions express commonsense notions that a pathway should stimulate its own output more than another pathway's output, and respond to its own input more than to another's. Moreover, they are simple ratios of pathway output, a property that is readily measurable by modeling or experiment. We also defined the informative metric of network specificity, the product of pathway specificities or fidelities. We demonstrated the utility of these metrics by calculating them for simple networks that share components, revealing the limited specificity inherent in simple architectures devoid of specificity‐promoting enhancements. Finally, we showed how the insulating mechanisms of compartmentalization and scaffolding are related, and how both require slow exchange rates and fast deactivation rates in order to promote high levels of specificity.
This work was supported by a National Institutes of Health/National Science Foundation joint initiative on Mathematical Biology through National Institute of General Medical Sciences grant GM75309 (LB, NK, QN), by NIGMS grant P20‐GM66051 (LB, QN), by a grant from the National Academies of Sciences Keck Futures Initiative and by NIGMS research grants GM60366 and GM69013 (LB) and by NIGMS grant GM67247 (QN). XZ is supported by Chinese National Natural Science Foundation grant no. 60573168.
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