Detection of long duration signals
In the model as previously developed[14], an input signal, f(t), activates the first member of the pathway whose activation can then activate the next member. In turn, each upstream species activates its immediate downstream target and can also be deactivated by, for example, a phosphatase. Assuming Michaelis Menten kinetics for the activation and deactivation of each species along the cascade, we can write an equation for the dynamics of the active form of the ithspecies x
i
along the cascade:
(1)
where E
pase
is the concentration of the enzyme that deactivates species i, , are the catalytic constants of the activation and deactivation steps, , are the Michaelis constants and is the total amount species available at step i. We take , at each step to be large (, > ) so that the kinetics of the reactions are not limited by the availability of the enzyme [21]. This assumption implies that the enzyme kinetics operate in a linear, first-order regime. Next, we assume that the cascade is weakly activated (i.e. at each stage, the total number of species is much larger than the number of active species, >> x
i
). In many biologically relevant instances (e.g. the Mitogen activated protein kinase (MAPK) cascade), the neglect of saturation effects is often reasonable [22]. Further, modeling the deactivation as a first order reaction is often valid when phosphatases are in excess as is the case in many physiological scenarios[23]. Eq. 1 simplifies to a system of linear first order differential equations[14]:
(2)
where the first species is activated at a rate f(t); and . This scheme is depicted in Fig. 2a.
The weakly activated cascade model has the advantage that the linearity of the equations allows for analytical tractability. Eq. 2 can be conveniently analyzed by introducing Fourier transformed variables: X
i
(ω) = ∫dteiωtx
i
(t) and F(ω) = ∫dteiωtf(t). The number of activated species at stage i becomes . The power spectrum P
i
(ω) ≡ |X
i
(ω)|2 at the ithstep can also be obtained: . After iterating at each successive stage of the n step cascade, an expression for P
n
(ω) as a function of the power spectrum of the input signal (S(ω) ≡ |F(ω)|2) is obtained:
P
n
(ω) = g
n
(ω)S(ω),
in which a frequency dependent gain g
n
(ω) is defined as:
(4)
g
n
(ω) is a transfer function that converts the input S(ω) into a response and provides a measure of the signal processing capabilities of the network. The change in the amplitude of the signal output is determined by the term and the time dependence of the output is modulated by the term.
From the formula of g
n
(ω), one consequence of having multiple stages is readily apparent. In the high frequency regime , for each , g
n
(ω) rapidly decays with increasing n (g
n
(ω) ~ ω-2n). Thus, longer cascades are more efficient at filtering the high frequency components of the signal from the output. This behavior is illustrated in Fig. 2b. Fig. 2b contains plots of g
n
(ω) for different values of n; cascades of lengths n = 1, 2, 3, 4 are shown.
In eq. 4, the relative values of along different stages of the cascade also affects the scaling behavior of g
n
(ω) as ω changes as well as the overall amplitude. The change in signal amplitude at the steady state (that leads to amplification or attenuation) at step i is given by the ratio of the effective rate constants for activation and deactivation . results in signal amplification and leads to attenuation of the signal at step i [13, 14]. Amplification or attenuation also leads to different time dependent behaviors of g
n
(ω). For example, consider an n staged cascade with rate constants such that , at frequencies ω > , g
n
(ω) ~ ω-2nwhile at frequencies > ω > , g
n
(ω) ~ ω-2(n-1)and so forth. Thus, at intermediate frequencies, a time scale separation (as determined by different deactivation rates along the cascade), along with signal amplification and attenuation, also leads to different frequency dependent behaviors of g
n
(ω).
Also, from the plots in Fig. 2c (and inspection of eq. 4), it is observed that incorporation of faster steps along the cascade influences the frequency dependence of g
n
(ω) to a lesser extent than would be the case when the kinetics of activation are same for each successive step. Since signal propagation in these cases is limited by the slower stages of the cascade, the faster steps are effectively removed from g
n
(ω). This observation suggests a principle in the ability of a biochemical pathway to filter signals of short duration: when there is a positive gradient of deactivation rates (that also leads to amplification or attenuation of signal amplitude), the time dependence of signal integration for multi staged cascades more closely resembles that of a pathway involving a single step. This effect is a result of a single dominant time scale in the pathway and provides a mechanism for regulating the amplitude of the signal output while keeping the time dependence of the output the same as that of a single-staged pathway.
Detection of short duration signals
While the sequential activation of multiple steps in a biochemical pathway allows for effective filtering of the high frequency, short duration components of a signal, often the desired signal output is regulated by feedback. We will now show with our analysis that feedback control in some instances also allows for the filtering of the low frequency, long duration components of a signal. Previous work has characterized this behavior with numerical simulations [17, 18]. In these instances, signals that occur at short times can be integrated while signals with a longer duration are effectively filtered because at longer times, the negative feedback loop affects the signal output.
For instance, the signal output can be affected by a feedback loop that is initiated downstream of the output. This scenario would be the case when the signal output from a biochemical cascade activates its own positive or negative regulators. For instance, in mammalian cells, the activation of extracellular regulatory kinase (ERK) often leads to the upregulation or activation of its own phosphatases[24]. In this scenario, a signal output in the form of phosphorylated ERK (ERK is known to phosphorylate on the order of one hundred substrates) is deactivated as a result of the upregulation of phosphatases that dephosphosphorylate residues in the TxY motif whose phosphorylation is necessary for activation.
Signal output is the activity of the kinase at the mthstep and feedback control to the signal output at step m is initiated at a later step (i.e. n > m) and the modified set of dynamical equations becomes:
(5)
where kfis the feedback strength and sets the time scale of the feedback and . This scheme is depicted in Fig. 3a.
Another biologically important example involves a negative feedback loop that acts upon a downstream layer in the cascade. However, in the linear cascade approximation that is used in this paper (that allows for extensive mathematical analysis) such a feedback loop would simply act to effectively increase the rate of deactivation of the species involved in the layer of the cascade that is involved in the feedback interaction. A model with nonlinear negative feedback (and one that is not analytically tractable) would be required to show the effect mathematically.
After applying a Fourier transformation as before, an expression for from eq. 5, albeit now more complicated, can be obtained as a function of S(ω) in closed-form:
(6)
where,
(7)
In the case of feedback regulation, there exists a competition between processes that are realized on multiple time scales: one for the signal to propagate along the cascade to the species involved in the signal output, and the others for the additional interactions derived from the feedback loops to propagate and interact with the species involved in the output. The competition between these effects in principle may lead to a frequency dependent optimal value of . At high frequencies as before, signal propagation is limited by the time it takes to move through the cascade and high frequency components of are filtered. Also, at low frequencies, signals can potentially be attenuated when the response is dominated by the activity of the feedback loop. If the interaction from the feedback is sufficiently strong, then the low frequency components of the signal are also filtered by the cascade. In this scenario, the frequency dependent behavior of would be non monotonic.
We illustrate these ideas through consideration of a three tiered cascade that consists of a chemical species carrying the input signal, a species conferring the signal output, and a species activated downstream to the output that provides a feedback interaction to the species conferring the signal output. In this scheme, m = 1 and n = 2, and eq. 7 is simplified and becomes:
(8)
The optimal frequency ω
opt
is obtained by differentiating ,
(9)
ω
opt
increases monotonically for decreasing values of and increasing values of k
f
. For negative feedback υ = -1, ω
opt
exists (Im ω
opt
= 0) when . Positive feedback υ = +1 requires that two conditions are satisfied for ω
opt
to be real, and . The height at the optimal frequency is:
(10)
We can also compute the width ω1/2 of at half maximum . ω1/2 has the form:
(11)
Where , , , and . Fig. 3b considers plots of for different feedback strengths k
f
. The curves in Fig. 3b illustrate changes in , , and ω1/2. Fig. 3c illustrates how , , and ω1/2 change for different values of .
Differential detection of long and short signals by two interacting species
In the previous sections, we illustrated how long and short duration signals can be differentially detected with different network structures. Alternative schemes for detecting long and short duration signals are also possible. Consider a mechanism that illustrates the time dependence of signal transduction involving the competition of two interacting products. One product is produced in greater amounts at one time scale and the other is produced in great amounts at a different time scale. Similar schemes have been investigated in the context of MAPK signaling and have shown to have many effects on integrated signal output[17, 18, 24, 25]. For example, in Yeast MAPK pathways, the output of one pathway (e.g. the stress-induced MAPK HOG1 pathway), can inhibit the activity of the (mating response-induced MAPK FUS3 pathway) as has been previously shown [25].
Here, we focus on the dynamics and time scale dependence of these mechanisms. Consider the following scheme depicted in Fig. 4a. In this scenario, two interacting products X and Y are produced by the same signal such as is the case of two parallel, interacting MAPK pathways. A set of two kinetic equations for species X and Y (denoted by x and y respectively) can be written as follows:
(12)
γ is the strength of the negative interaction from species Y to species X. The other parameters that are introduced in this model are α1 and α2, that set the strength of interaction between the signal stimulus f(t) and X and Y respectively. The constitutive degradation rate constants of X and Y are β1 and β2.
We can solve eq. 12 using Fourier transformation as before. We define the following as before: P
X
(ω) ≡ |X(ω)|2 and P
Y
(ω) ≡ |Y(ω)|2; where, and , and S(ω) = |F(ω)|2 and :
P
X
(ω) = g
X
(ω)S(ω)
and
P
Y
(ω) = g
Y
(ω)S(ω).
The frequency dependent gain for species X and Y are obtained,
(14)
and
Plots of g
X
(ω) (solid) and g
Y
(ω) (dashed) are considered in Fig. 4b. From Fig. 4b, the following behavior is apparent. At very high frequencies, signals from f(t) are filtered by both products X and Y. At intermediate frequencies (ω ~ 1.0 to ω ~ 10.0) signals deriving from f(t) are integrated more efficiently by species X. Therefore, short duration signals are integrated more efficiently by species X. At low frequencies, ω < 1.0, signals originating from f(t) are integrated more efficiently by species Y. As a result, long duration signals are integrated more efficiently by species Y since at long times, X is affected by the negative interaction from Y.
This simple two species model illustrates how signal specificity can be achieved from two competing products by introducing changes in signal duration of the upstream signal. Short duration signals are more effectively integrated by one species and long duration signals are more effectively detected by the other.
Integration of signals of differing duration
In the previous sections, we considered the frequency dependent gain of different network structures. In this section, we consider an incoming signal of differing duration and observe how it is differentially processed by networks that filter signals at different time scales. First, we considered the case in which the network filters short duration (high frequency) signals (Fig. 2). Next, we considered the case (Fig. 3) in which the network filters signals of long duration (low frequency).
A convenient way to parameterize signals of differing duration, while keeping the total amount of signal ∫ f(t)dt = α fixed, is to consider the function,
(15)
where Θ(t) is a Heaviside step function, τd sets the signal duration, and α is taken to be 1 (α = 1) in the appropriate units. This form of f(t) models the behavior of a typical experimental signaling time course [4]. Plots of f(t) are shown in Fig. 5a. For this choice of signal, S(ω) is easily computed;
(16)
S(ω) is plotted in Fig. 5b for different values of τd ranging from = 2.0 (short duration) to = 0.1 (long duration).
Figs. 5c and 5d illustrate how signals S(ω) of large ( = 0.5 dotted lines) and small ( = 2.0 dashed lines) duration are integrated by the internal gains g3(ω) (Fig. 5c, from eq. 4) and (Fig. 5d, from eq. 8) of these multistage cascades of differing network topologies. In Fig. 5c, the signal output P3(ω) (from eq. 3), upon integration by a three-tiered kinase cascade is shown. Taking for i ∈ 1, 2, 3, g3(ω) effectively filters the short ( = 2.0) duration signal and results in an output P3(ω) of small magnitude at all time scales 2πω-1 in the frequency spectrum. In contrast, for the signal characterized by = 0.5, signal processing through g3(ω) results in a signal of larger amplitude. The ratio of amplitudes (with the superscript denoting the duration used) at the optimal frequency (ω = 0) for the two signals is ≈ 17.
In Fig. 5d, the signal output (from eq. 6), obtained from a signal output that is also affected by a downstream negative (υ = -1) feedback loop, is shown. Parameters used are: = 2.0, = 1.0, = 1.0, = 0.01, kf= -5.0. For the signal of long duration = 0.5, only the small frequency components of the signal are integrated. This behavior is in contrast with the signal output of a short duration signal = 2.0. The amplitude difference in this case is ≈ 0.2.