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 Xout|Xin, 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.
These measures of output under different pathways inputs are used to express the metrics specificity and fidelity[22, 25]. A pathway is said to have output specificity if that pathway's input activates its own output more than it does the output of any interconnected pathway. A pathway is said to have input fidelity if the output is greater when it receives its own signal than it is when it receives an interconnected pathway's signal. These two concepts can be quantified as:
(1.1)
where S
X
denotes the output specificity in the X pathway and F
Y
denotes the input fidelity in the Y pathway, etc.
In order to escape obvious logical contradictions and function effectively, a signaling network needs to posses output specificity and input fidelity all its pathways simultaneously. To account for this in the context of a two-pathway network, we define three composite indicators, the degree of Mutual Fidelity (MF), Mutual Specificity (MS) and Mutual Fidelity & Mutual Specificity (MFMS):
(1.2)
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
The scheme depicted in Fig. 1A can be modeled as a system of ordinary differential equations:
(2.1)
(2.2)
(2.3)
These equations describe the formation of active signaling species x1, x2 and y2, and do not explicitly consider the inactive precursors from which they are converted. The parameters a1 and a2 are activation rate coefficients; a2 is proportional to the rate at which component x1 activates 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. The functions fXand fYare activation functions that describe how the rate of change with respect to time of x2 and y2 vary as a function of the concentration of active x1. For weakly-activated signaling pathways (i.e. pathways in which, at physiological levels of input, only a small fraction of any given component becomes activated), the production of x2 and y2 is a linear function of x1
In contrast, for hyperbolic pathways, and for ultrasensitive pathways, the activation functions fX and fY can often be reasonably approximated by Hill functions:
(2.4)
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
We will use the notation to refer to the steady state value of x1 given that pathway X is on (that is, activated) and Y is off; we could also have written x1|X
in
. Similarly, refers to the steady state value of x1 when X is off and Y is on. As stated above, for weakly-activated pathways, activation kinetics are linear [23, 24], and so , and . If we define the quantities and (where α measures the connection strength from x1 to x2, and β from x1 to y2), we can then express the values of S
X
and S
Y
for the weakly-activated system simply as
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].
When pathways are hyperbolic or ultrasensitive, mutual specificity becomes possible, even in the basic architecture. In these cases, the equations for S
X
and S
Y
are:
(3.1)
For hyperbolic but not ultrasensitive pathways, n = m = 1, and eqs. (3.1) reduce to
(3.2)
S
X
can be made large by setting α/β > > 1 and letting , whereas S
Y
can be made large by setting (ε
X
/ ε
Y
) > > 1 and letting . In this case S
X
→ α/β and S
Y
→ (β · ε
X
)/(α · ε
Y
), which will both be greater than one so long as
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
If we assume that the activation functions fX and fY are monotonic, but make no other assumptions as to their specific form, we can readily prove that mutual fidelity is impossible at steady state in the absence of an insulating mechanism. Let us consider the steady state of the system (see Fig. 2B and 2C for an illustration of the analysis below). Clearly x1 must take on different values given either X input or Y input, otherwise neither X nor Y fidelity would be possible at steady state. Suppose that . As the functions fX and fY are activation functions, they are assumed to be monotonic and increasing, therefore more x1 gives more x2 and more y2. (We are assuming no other structure on the activation functions other than the fact that they are monotonic, therefore this result holds regardless of whether the functions are linear, Hill-like, or any other always-increasing function.). So if , then it must be that the steady state value of x2 given X input, X
out
|X
in
, must be greater than x2 given Y input, that is X
out
|X
in
> X
out
|Y
in
. This is, in fact, the definition of fidelity in the X pathway
(4.1)
Thus, fidelity in the X pathway is guaranteed. However, this same argument also implies that Y
out
|X
in
> Y
out
|Y
in
. This is exactly the statement that fidelity in the Y pathway
(4.2)
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 , 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, fX and fY . 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.
Cross-pathway inhibition occurs when one pathway inhibits another pathway. Here we consider a particular implementation of this, where both the inhibiting and inhibited components are downstream of a shared branchpoint (Fig. 1B). In the yeast MAPK network, both the MAP kinase Fus3 (an output specific to the mating pathway) and the transcription factor Tec1 (an output specific to the filamentation pathway) are downstream of the shared kinase cascade. Tec1 activation during mating is prevented, in part, because Fus3 phosphorylates Tec1 and thereby targets Tec1 for ubiquitin-mediated degradation [46–48]. Other likely examples of this type of cross-pathway inhibition in the yeast MAPK network include inhibition of Tec1 by the stress-response kinase Hog1 [49], and inhibition of Hog1 by the filamentation kinase Kss1 [37]. Following [22, 25], we incorporate insulating mechanisms into the system composed of Eqs. (2.1)(2.2)(2.3). In cross-pathway inhibition, the equation for y 2, (2.3), becomes
(5.1)
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.
In combinatorial signaling, in order for input from the X pathway to evoke a response, an independent input from a third receptor (Z) is required (see Fig. 1C). The component x2 acts a coincidence detector that only responds if both x1 and Z are active. In this case, the equation for x2, (2.2), becomes
(5.2)
where
(5.3)
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] ≡ kleak, where kleak, 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 kleak compared to the basic architecture. Hence, kleak = 1 has no specificity enhancing effect, whereas kleak = 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].
The sequestering function of scaffolds is implemented by expanding the system to include two different states of the "shared" component: bound to the scaffold (denoted , for aNchored to the scaffold), and free in the cytosol (denoted ). It is presumed that active x2 can only be created by x1 that is bound to the scaffold, and that x1 bound to scaffold cannot create active y2. That is, X pathway output is a function of and the Y pathway output is a function of , as shown below:
(5.4)
(5.5)
(5.6)
(5.7)
The same set of equations can be used to describe the insulation mechanism of compartmentalization[22]. In compartmentalization, the X pathway is presumed to reside in one cellular compartment (e.g. the nucleus) and the Y pathway to reside in another (e.g. the cytosol). Leaking between the pathways can occur because the shared component can move between these two compartments to some extent. For instance, some portion of the pool of x1 activated in the nucleus () may move into the cytosol (becoming ), giving it the opportunity to inappropriately create y2. Thus, we refer to the insulating mechanism modeled by Eqns (5.4)-(5.7) as scaffolding/compartmentalization (SC). SC works by creating two different states for the shared component. These states are allowed to freely transform between one another:
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.
Cross Regulatory Term (CRT)
In the following sections we will compare the effect of each of the above insulating mechanisms on the signaling pathway's ability to achieve MFMS, both numerically and analytically. In many cases one can show that arbitrarily high degrees of MFMS can be achieved at steady state. In other words for any k there is a set of parameters so that MFMS > k. However the realization of increasingly high degrees of MFMS requires more and more extreme choices of parameters and increasing cross-regulation. Therefore we need to be able to quantify the degree of additional regulation attributable to the insulating mechanism. Thus, for each of the different insulating mechanisms defined above, we identified a key dimensionless parameter to quantify the degree of additional regulation. We call this the Cross Regulatory Term (CRT); it is defined as follows:
(5.8)
where for SC we let D
in
= D
out
≡ D and .
Each of the CRTs were chosen intuitively as a set of parameters that quantifies the cross pathway regulation. For example with combinatorial signaling, the leak rate is clearly the parameter that quantifies the cross pathway regulation, as it is the only parameter that differentiates a CS network from the basic architecture. Both numerical (data not shown) and analytic results (see below) show that the CRTs as defined are in fact critical for determining specificity.
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.
Upon inspection F
Y
> 1 only when , which then makes F
X
< 1. Therefore, regardless of the CRT, MFMS ≤ 1.
In the case of combinatorial signaling, however, one can show that . In this case
The maximum of this expression over all of the parameters occurs when , and at this point .
For scaffolding/compartmentalization (SC), the output specificity and input fidelity readily are calculated:
Evaluating these expressions, we find:
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).
Therefore we can assert
(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 and therefore and thus
or and therefore and thus
Thus in any case we have,
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.
(6.3)
we obtain the bound
(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.
If ultrasensitivity is helpful to MFMS, then increasing n and m should increase S
X
, S
Y
, F
X
and F
Y
. (See section 4 of Additional file 1 for derivation. This same approach can be taken on the steady states of x2 and y2 directly with the same result we derive below). First observe the result in the case where there are no insulating mechanisms. Taking derivatives:
(7.1)
Combining these conditions gives,
(7.2)
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 or .
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:. 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: . 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.
In marked contrast to the above, scaffolding/compartmentalization allows for all four equations to be simultaneously satisfied. SC creates two distinct species of the shared component and therefore the derivatives with respect to Hill Exponents change to:
(7.3)
Combining these new equations gives:
(7.4)
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 and the Y pathway is only activated by one can set the threshold for 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 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]).
To do this we use as a characteristic value the steady state value of X
out
|X
in
for the X pathway and Y
out
|Y
in
for the Y pathway and we define new normalized variables, denoted :
With these new definitions of normalized variables we calculate the output specificity
and
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.