Signal Duration and the Time Scale Dependence of Signal Integration in Biochemical Pathways

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Citation Locasale, Jason W. 2008. Signal duration and the time scale dependence of signal integration in biochemical pathways. BMC Systems Biology 2:108.

Published Version doi:10.1186/1752-0509-2-108

Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:10236190

Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA BMC Systems Biology BioMed Central

Research article Open Access Signal duration and the time scale dependence of signal integration in biochemical pathways Jason W Locasale1,2

Address: 1Department of Biological Engineering, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139, USA and 2Department of Systems Biology, Harvard Medical School, Division of Signal Transduction, Beth Israel Deaconess Medical Center, Boston MA 02115, USA Email: Jason W Locasale - [email protected]

Published: 17 December 2008 Received: 23 June 2008 Accepted: 17 December 2008 BMC Systems Biology 2008, 2:108 doi:10.1186/1752-0509-2-108 This article is available from: http://www.biomedcentral.com/1752-0509/2/108 © 2008 Locasale; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract Background: Signal duration (e.g. the time over which an active signaling intermediate persists) is a key regulator of biological decisions in myriad contexts such as cell growth, proliferation, and developmental lineage commitments. Accompanying differences in signal duration are numerous downstream biological processes that require multiple steps of biochemical regulation. Results: Here we present an analysis that investigates how simple biochemical motifs that involve multiple stages of regulation can be constructed to differentially process signals that persist at different time scales. We compute the dynamic, frequency dependent gain within these networks and resulting power spectra to better understand how biochemical networks can integrate signals at different time scales. We identify topological features of these networks that allow for different frequency dependent signal processing properties. Conclusion: We show that multi-staged cascades are effective in integrating signals of long duration whereas multi-staged cascades that operate in the presence of negative feedback are effective in integrating signals of short duration. Our studies suggest principles for why signal duration in connection with multiple steps of downstream regulation is a ubiquitous motif in biochemical systems.

Background topology that collectively integrate an incoming signal to Signal duration (e.g. the length of time over which a sign- deliver a specific biological response. The sequential acti- aling intermediate is active) is a critical determinant in vation of multiple steps in a biochemical pathway is a mediating cell decisions in numerous biological processes ubiquitous regulatory motif involved in many aspects of including cell growth, proliferation, and developmental gene regulation, metabolism, and intracellular signal lineage commitments (Fig. 1) [1-8]. One fundamental transduction. Many advantages of having multiple steps issue in signal transduction and cell decision making then of regulation as opposed to having activation occur is how differences in signal duration are detected to through a single step have been documented. A signaling achieve the appropriate biological response. cascade can allow for attenuation of noise, incorporation of additional regulatory checkpoints or proofreading Accompanying changes in signal duration are multiple steps, and increased tunability of the input signal [9-12]. stages of biochemical regulation of differing network Other studies have established conditions under which

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Signal Duration Short Long

Physiological Process

Adipogenesis Adipogenesis Proliferation

Thymic Selection in Negative Selection (Apoptosis) Positive Selection T Cell Development

Neuronal Stem Cell ---- Proliferation Differention

Cytokine Signaling Inflammatory Response Anti-Inflammatory Response

PhysiologicalFigure 1 examples of signal duration determining the phenotypic outcome in signal transduction Physiological examples of signal duration determining the phenotypic outcome in signal transduction. Four examples of physiological processes in which branching phenotypic decisions are believed to be controlled by the detection of differences in signal duration [5,6,8,28-30].

signaling cascades amplify or attenuate incoming signals matic activation of multiple species along a pathway to [13-15]. These conditions are established by rates of acti- understand mechanistic principles underlying how multi- vation, rates of deactivation that are set by phosphatase ple stages in a biochemical pathway can integrate differ- activities, and the presence of scaffold proteins [14-16]. ences in signal duration. We use a model of a weakly However, how these features synergize with downstream activated cascade[13,14,19], whose assumptions we first effector pathways to detect differences in signal duration motivate, to study the question of how biochemical cas- has not been fully studied. cades detect the time scale dependence of input signals. Our approach is similar to previous work [20] that inves- A recent study has proposed a model that predicts how tigated the frequency dependent signal processing proper- signals with different dynamical characteristics can be dis- ties of single enzymatic cycles. The model allows us to tinguished upon integration into different network archi- characterize the dynamics by obtaining exact expressions tectures [17,18]. We develop a formalism to complement for the power spectra of linearized biochemical networks their approach and, as a consequence, identify general of multiple stages with arbitrary length and connectivity principles for how different network topologies can differ- and we focus on how these frequency dependent signal entially integrate signals that persist at different time processing properties of cascades are used to detect differ- scales. We focus on a simple model of the sequential enzy- ences in signal duration.

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We first show that biochemical cascades can function as excess as is the case in many physiological scenarios[23]. both high low and high pass filters depending on the Eq. 1 simplifies to a system of linear first order differential topology of the network architecture. A low pass filter equations[14]: removes high frequency (short duration) components of a signal and a high pass filter removes low frequency (long dx +− duration) components of a signal. These filtering capabil- 1 = kf() t− kx dt 111 ities are determined by differential positive and negative , (2) regulation within the biochemical pathway. Importantly, dxi =−+ − kxii−1 kx ii the filtering capabilities are determined by the presence of dt feedback as well as the amplification and attenuation + properties at different steps in the cascade that are set by where the first species is activated at a rate k1 f(t); the differences in phosphatase activities at different stages T i i + x k − E k along the cascade. Ultimately, our findings suggest design k = i cat and k = pase cat, pase . This scheme is i K Mi, i i principles that characterize how biochemical cascades are K Mpase, well suited for detecting time scale dependent differences depicted in Fig. 2a. in biochemical signals. The weakly activated cascade model has the advantage Results that the linearity of the equations allows for analytical Detection of long duration signals In the model as previously developed[14], an input sig- tractability. Eq. 2 can be conveniently analyzed by intro- iωt nal, f(t), activates the first member of the pathway whose ducing Fourier transformed variables: Xi(ω) = ∫dte xi(t) activation can then activate the next member. In turn, and F(ω) = ∫dteiωt f(t). The number of activated species at each upstream species activates its immediate down- + () stream target and can also be deactivated by, for example, ()= ki Xi−1 ω stage i becomes X i ω − . The power spec- a phosphatase. Assuming Michaelis Menten kinetics for ikω + i the activation and deactivation of each species along the trum P (ω) ≡ |X (ω)|2 at the ith step can also be obtained: cascade, we can write an equation for the dynamics of the i i th + 2 active form of the i species xi along the cascade: ()ki PP()ωω= − (). After iterating at each suc- ii− 2 1 ω 2 +()k ki x − xT −x i i dx cat i 1()i i kcat, paseEpasexi i = − , (1) dt i T i + cessive stage of the n step cascade, an expression for Pn(ω) K +−()x xi K xi M i Mpa, sse as a function of the power spectrum of the input signal (S(ω) ≡ |F(ω)|2) is obtained: where Epase is the concentration of the enzyme that deacti- i i Pn(ω) = gn(ω)S(ω), (3) vates species i, kcat , kcat, pase are the catalytic constants of

i i the activation and deactivation steps, k M , k Mpase, are the in which a frequency dependent gain gn(ω) is defined as:

T Michaelis constants and xi is the total amount species ⎛ + ⎞2 n k i i = ⎜ i ⎟ 1 available at step i. We take k , k at each step to be g n ()ω ∏ . M Mpase, ⎜ − ⎟ 2 (4) i=1 ⎝ ki ⎠ ⎛ ⎞ i i T +⎜ ω ⎟ large (k M , k Mpase, > xi ) so that the kinetics of the reac- 1 ⎜ − ⎟ ⎝ ki ⎠ tions are not limited by the availability of the enzyme [21]. This assumption implies that the enzyme kinetics gn(ω) is a transfer function that converts the input S(ω) operate in a linear, first-order regime. Next, we assume into a response and provides a measure of the signal that the cascade is weakly activated (i.e. at each stage, the processing capabilities of the network. The change in the total number of species is much larger than the number of amplitude of the signal output is determined by the T active species, xi >> xi). 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

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FilteringFigure 2of high frequency signals Filtering of high frequency signals. Time dependence of signal integration in a linear biochemical cascade. a.) the sequential activation of multiple stages in a signaling cascade. Superscripts (I) and (A) denote inactive and active forms of each chemical species and are dropped from the equations in the text. b.) same kinetic constants, all kinetic constants are taken to be: + k +−== i = + + + kkii10. c.) a positive gradient of activation/deactivation rates keeping − 10. fixed. k1 = 1.0, k2 = 3.3, k3 = 6.6, ki + k + − i = − k4 = 10.0. c.) plots of gn(ω) for n = 1, 2, 3, 4 with successively different values of ki while keeping − 10. fixed (k1 = 1.0, ki − − − k2 = 3.3, k3 = 6.6, k4 = 10.0).

2 n ⎛ + ⎞ 2b. Fig. 2b contains plots of gn(ω) for different values of n; ⎜ ki ⎟ ∏ ⎜ − ⎟ term and the time dependence of the output is cascades of lengths n = 1, 2, 3, 4 are shown. i=1 ⎝ ki ⎠ −1 2 − n ⎡ ⎛ ⎞ ⎤ In eq. 4, the relative values of k along different stages of ⎢ ⎥ i modulated by the ∏ 1 + ⎜ ω ⎟ term. ⎢ ⎜ − ⎟ ⎥ the cascade also affects the scaling behavior of g (ω) as ω i=1 ⎣ ⎝ ki ⎠ ⎦ n changes as well as the overall amplitude. The change in signal amplitude at the steady state (that leads to amplifi- From the formula of g (ω), one consequence of having n cation or attenuation) at step i is given by the ratio of the multiple stages is readily apparent. In the high frequency effective rate constants for activation and deactivation ⎛ ⎞ + + + ⎜ ω >> ⎟ − k k k regime − 1 , for each ki , gn(ω) rapidly decays with i . i > 1 results in signal amplification and i < 1 ⎝ k ⎠ − − − i ki ki ki -2n increasing n (gn(ω) ~ ω ). Thus, longer cascades are more leads to attenuation of the signal at step i [13,14]. Ampli- efficient at filtering the high frequency components of the fication or attenuation also leads to different time signal from the output. This behavior is illustrated in Fig.

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on the order of one hundred substrates) is deactivated as dependent behaviors of gn(ω). For example, consider an n − a result of the upregulation of phosphatases that dephos- staged cascade with rate constants ki such that phosphorylate residues in the TxY motif whose phospho- − − −− − >> > - rylation is necessary for activation. kknn−121 k n− ... k, at frequencies ω > kn , gn(ω) ~ ω − − 2n -2(n-1) th while at frequencies kn > ω > kn−1 , gn(ω) ~ ω and Signal output is the activity of the kinase at the m step so forth. Thus, at intermediate frequencies, a time scale and feedback control to the signal output at step m is ini- tiated at a later step (i.e. n > m) and the modified set of separation (as determined by different deactivation rates dynamical equations becomes: along the cascade), along with signal amplification and attenuation, also leads to different frequency dependent dx1 = +−()− behaviors of g (ω). kf111 t kx n dt (5) dxm + − f =−+kx− kxυ kx Also, from the plots in Fig. 2c (and inspection of eq. 4), it dt mm1 mm n is observed that incorporation of faster steps along the cas- cade influences the frequency dependence of gn(ω) to a where kf is the feedback strength and sets the time scale of lesser extent than would be the case when the kinetics of ⎧ 1; positive feedback ⎫ activation are same for each successive step. Since signal the feedback and υ = ⎨ ⎬ . This − propagation in these cases is limited by the slower stages ⎩ 1;negative feedback ⎭ of the cascade, the faster steps are effectively removed scheme is depicted in Fig. 3a. from gn(ω). This observation suggests a principle in the ability of a biochemical pathway to filter signals of short Another biologically important example involves a nega- duration: when there is a positive gradient of deactivation tive feedback loop that acts upon a downstream layer in rates (that also leads to amplification or attenuation of the cascade. However, in the linear cascade approxima- signal amplitude), the time dependence of signal integra- tion that is used in this paper (that allows for extensive tion for multi staged cascades more closely resembles that mathematical analysis) such a feedback loop would sim- of a pathway involving a single step. This effect is a result ply act to effectively increase the rate of deactivation of the of a single dominant time scale in the pathway and pro- species involved in the layer of the cascade that is involved vides a mechanism for regulating the amplitude of the sig- in the feedback interaction. A model with nonlinear neg- nal output while keeping the time dependence of the ative feedback (and one that is not analytically tractable) output the same as that of a single-staged pathway. would be required to show the effect mathematically.

Detection of short duration signals After applying a Fourier transformation as before, an While the sequential activation of multiple steps in a bio- f ω chemical pathway allows for effective filtering of the high expression for Pm() from eq. 5, albeit now more compli- frequency, short duration components of a signal, often cated, can be obtained as a function of S(ω) in closed- the desired signal output is regulated by feedback. We will form: now show with our analysis that feedback control in some instances also allows for the filtering of the low frequency, PgSf ()ωωω= f ()(), (6) long duration components of a signal. Previous work has m m characterized this behavior with numerical simulations where, [17,18]. In these instances, signals that occur at short + 2 times can be integrated while signals with a longer dura- m ()k f = i gm ()ω ∏ ⎛ − 2 ⎞ tion are effectively filtered because at longer times, the i=1 ⎜ + 2 ⎟ ⎜ ()ki ω ⎟ negative feedback loop affects the signal output. ⎝ ⎠ 22⎡ n + ⎤ ()k f ⎢ ∏ ()k ⎥ ⎢ i ⎥ + ⎣ i=1 ⎦ ⎛ − 2 ⎞ n ⎛ − 2 ⎞ 22n + m−1⎛ − 2 ⎞ n + 22m−1⎛ − ⎞ For instance, the signal output can be affected by a feed- ⎜ + 2 ⎟ ∏ ⎜ + 2 ⎟+ f ∏ ∏ ⎜ + 2 ⎟+ ∏ ∏ ⎜ + 2 ⎟ ⎜ ()km ωω⎟ ⎜ ()ki ⎟ ()k ()ki ⎜ ()ki ωω⎟ 2 ()ki ⎜ ()ki ⎟ back loop that is initiated downstream of the output. This ⎝ ⎠i=1⎝ ⎠ im=+1 i=1 ⎝ ⎠ im= i=1 ⎝ ⎠ ⎡ n + ⎤⎡ m + ⎤ k f ⎢ ∏ ()k ⎥⎢ ∏ ()k ⎥ scenario would be the case when the signal output from a ⎢ i ⎥⎢ i ⎥ + υ ⎣ i=11⎦⎣ i= ⎦ 2 ⎡ n − ⎤ m ⎛ − 2 ⎞ ⎛ − 2 ⎞ n + m−1⎛ − 2 ⎞ biochemical cascade activates its own positive or negative ⎢ ∏ ⎥ ∏ ⎜ + 2 ⎟− f ⎜ + 2 ⎟ ∏ ∏ ⎜ + 2 ⎟ ()ki ⎜ ()ki ωυ⎟ k ⎜ ()km ω⎟ ()ki ⎜ ()ki ω ⎟ regulators. For instance, in mammalian cells, the activa- ⎣⎢ im= ⎦⎥i=1⎝ ⎠ ⎝ ⎠im=+1 i=1 ⎝ ⎠ tion of extracellular regulatory kinase (ERK) often leads to (7) the upregulation or activation of its own phos- In the case of feedback regulation, there exists a competi- phatases[24]. In this scenario, a signal output in the form tion between processes that are realized on multiple time of phosphorylated ERK (ERK is known to phosphorylate scales: one for the signal to propagate along the cascade to

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0 0 1 2 3 4 5 6 Z FilteringFigure 3of low frequency signals Filtering of low frequency signals. A three tiered biochemical cascade with competing processes occurring at different time scales that is sufficient to filter signals at long time scales (Low pass filtering). a.) An initial stimulus activates a downstream species that confers a signal output and activates a downstream species that, through feedback, interacts with the species that carries the signal output. Superscripts (I) and (A) denote inactive and active forms of each chemical species and are omitted +−+=== − from the equations in the text. b.) parameter values were taken to be: υ = -1, kkk11210. ; k2 = 0.1. kf = 2.5 (solid lines), kf = 1.0 (dashed lines), kf = 0.5 (dotted lines), kf = 0.1 (dash-dotted lines). c.) parameter values were taken to be: υ = -1, +−+==== − − − − kkkk112f 10. . k2 = 0.0 (solid lines), k2 = 0.5 (dashed lines), k2 = 1.0 (dotted lines), k2 = 2.0 (dash-dotted lines). the species involved in the signal output, and the others components of the signal are also filtered by the cascade. for the additional interactions derived from the feedback In this scenario, the frequency dependent behavior of loops to propagate and interact with the species involved f gm()ω would be non monotonic. in the output. The competition between these effects in principle may lead to a frequency dependent optimal We illustrate these ideas through consideration of a three f value of gm()ω . At high frequencies as before, signal tiered cascade that consists of a chemical species carrying propagation is limited by the time it takes to move the input signal, a species conferring the signal output, through the cascade and high frequency components of and a species activated downstream to the output that f provides a feedback interaction to the species conferring gm()ω are filtered. Also, at low frequencies, signals can the signal output. In this scheme, m = 1 and n = 2, and eq. potentially be attenuated when the response is dominated f by the activity of the feedback loop. If the interaction from 7 is simplified and g1 ()ω becomes: the feedback is sufficiently strong, then the low frequency

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f f and ω1/2. Fig. 3c illustrates how g1 ()ω , g1 ()ωopt , and ω1/ + 2 ⎡ − 2 ⎤ ⎢ + 2 ⎥ ()kk1 ()2 ω − ⎣⎢ ⎦⎥ 2 change for different values of k2 . g f ()ω = . 1 2 ⎛ 2 2 ⎞ −−− + + 2⎜ − + − + + + 2 ⎟ ()kk12υω kf kkk2 ⎜ ()1 ()2 2υωk f k2 ⎟ ⎝ ⎠ Differential detection of long and short signals by two interacting species (8) In the previous sections, we illustrated how long and short

The optimal frequency ωopt is obtained by differentiating duration signals can be differentially detected with differ- ent network structures. Alternative schemes for detecting f ω g1 (), long and short duration signals are also possible. Con- sider a mechanism that illustrates the time dependence of 12/ ⎡ 12/ 2 ⎤ signal transduction involving the competition of two =−+++ −−−− − ωυopt⎢{}kk f22( kk f 2 k 212() k k) () k2 ⎥ . interacting products. One product is produced in greater ⎣ ⎦ amounts at one time scale and the other is produced in (9) great amounts at a different time scale. Similar schemes − ω increases monotonically for decreasing values of k have been investigated in the context of MAPK signaling opt 2 and have shown to have many effects on integrated signal and increasing values of kf. For negative feedback υ = -1, output[17,18,24,25]. For example, in Yeast MAPK path- ways, the output of one pathway (e.g. the stress-induced ωopt exists (Im ωopt = 0) when MAPK HOG1 pathway), can inhibit the activity of the 14/ ⎡ ++−−−++⎤ > − (mating response-induced MAPK FUS3 pathway) as has ⎢ kkff22212{} kk2 k() k k⎥ k2 . Positive feed- ⎣ ⎦ been previously shown [25]. back υ = +1 requires that two conditions are satisfied for 14/ Here, we focus on the dynamics and time scale depend- ⎡ ++−−− ⎤ > − ωopt to be real, ⎣ kkff2212() kk2 kk⎦ k2 and ence of these mechanisms. Consider the following scheme depicted in Fig. 4a. In this scenario, two interact- +−−−>+ kkf 22122 k() k k . The height at the optimal fre- ing products X and Y are produced by the same signal such as is the case of two parallel, interacting MAPK pathways. f quency g1 ()ωopt is: A set of two kinetic equations for species X and Y (denoted by x and y respectively) can be written as follows: + 2 ()k f = 1 dx g1 ()ωopt . = ()−− − 2 − 2 ⎧ + ++⎡ −−− ⎤ ⎫ αβγ11ft x y kk− +2⎨ kkk+ kkkkkk−2 + ⎬ dt ()1 ()1 f 222f ⎣⎢ f 21() 2⎦⎥ (12) ⎩ ⎭ dy = αβft()− y (10) dt 22 f We can also compute the width ω1/2 of g1 ()ω at half max- γ is the strength of the negative interaction from species Y f to species X. The other parameters that are introduced in imum g1 ()/ωopt 2 . ω1/2 has the form: this model are α1 and α2, that set the strength of interac- tion between the signal stimulus f(t) and X and Y respec- 1 ⎡ ⎤ ω=+++4 γγ 822 γγγγ()+− 4 γγ +− 822 γγγγ() + + , 12/ 2 ⎣⎢ 12 1 1 3 4 12 1 1 3 4⎦⎥ tively. The constitutive degradation rate constants of X (11) and Y are β1 and β2.

++−−−⎡ ⎤ We can solve eq. 12 using Fourier transformation as Where γ =−+kk kk2 k() k k , 122212ff⎣ ⎦ 2 before. We define the following as before: PX(ω) ≡ |X(ω)| 2 2 2 = −−− + +=+ − ∞ γ 21()kkkk32()2 f 2 , γγ322()k 2, and 2 = itω and PY(ω) ≡ |Y(ω)| ; where, Xext()ω ∫ () and −∞ 2 2 =+−− −−− + + γ 412()kk() kk12 4 kkf 2 . Fig. 3b considers ∞ Yeyt()ω = itω (), and S(ω) = |F(ω)|2 and f ∫ plots of g1 ()ω for different feedback strengths kf. The −∞ ∞ curves in Fig. 3b illustrate changes in g f ()ω , g f ()ω , 1 1 opt Feft()ω = ∫ itω (): −∞

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AFigure model 4 of two interacting species produced by the same signal A model of two interacting species produced by the same signal. Differential time dependent signal detection by two competing products. a.) two species, denoted by X and Y in are produced by the same signal f(t). Species Y negatively interacts with species X. Species X can positively interact with itself. The activity of species f is transient with associated signal duration. b.) frequency dependent gain of two interacting products. Plots of gX(ω) (solid line) and gY(ω) (dashed line) are shown. All parameters are taken to be 1.0 (in appropriate units) with the exception of α1; α2 = 1.5.

PX(ω) = gX(ω)S(ω) (13) frequencies, ω < 1.0, signals originating from f(t) are inte- grated more efficiently by species Y. As a result, long dura- and tion signals are integrated more efficiently by species Y since at long times, X is affected by the negative interac- PY(ω) = gY(ω)S(ω). tion from Y.

The frequency dependent gain for species X and Y are This simple two species model illustrates how signal spe- obtained, cificity can be achieved from two competing products by introducing changes in signal duration of the upstream 2 ⎡ 2 ⎤ ()()αβωγαγααβ+ 2 +−()2 signal. Short duration signals are more effectively inte- 1 ⎣⎢ 2 ⎦⎥ 2212 g ()ω = grated by one species and long duration signals are more X ⎡ ⎤ ωω42+ () β2+() β2 +()ββ 2 effectively detected by the other. ⎣⎢ 1 2 ⎦⎥ 12 (14) Integration of signals of differing duration In the previous sections, we considered the frequency and dependent gain of different network structures. In this sec- tion, we consider an incoming signal of differing duration ()α 2 and observe how it is differentially processed by networks g ()ω = 2 . that filter signals at different time scales. First, we consid- Y 2 2 ωβ+()2 ered the case in which the network filters short duration (high frequency) signals (Fig. 2). Next, we considered the Plots of gX(ω) (solid) and gY(ω) (dashed) are considered case (Fig. 3) in which the network filters signals of long in Fig. 4b. From Fig. 4b, the following behavior is appar- duration (low frequency). ent. At very high frequencies, signals from f(t) are filtered by both products X and Y. At intermediate frequencies (ω A convenient way to parameterize signals of differing ~ 1.0 to ω ~ 10.0) signals deriving from f(t) are integrated duration, while keeping the total amount of signal ∫ f(t)dt more efficiently by species X. Therefore, short duration = α fixed, is to consider the function, signals are integrated more efficiently by species X. At low

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S(ω) is plotted in Fig. 5b for different values of τd ranging −t /τ te dΘ()t −1 −1 ft()= α , (15) from τ d = 2.0 (short duration) to τ d = 0.1 (long dura- −t /τ Θ ∫ te d ()tdt tion). where Θ(t) is a Heaviside step function, τ sets the signal − d Figs. 5c and 5d illustrate how signals S(ω) of large (τ 1 = duration, and α is taken to be 1 (α = 1) in the appropriate d units. This form of f(t) models the behavior of a typical −1 0.5 dotted lines) and small (τ d = 2.0 dashed lines) dura- experimental signaling time course [4]. Plots of f(t) are shown in Fig. 5a. For this choice of signal, S(ω) is easily tion are integrated by the internal gains g3(ω) (Fig. 5c, computed; f from eq. 4) and g1 ()ω (Fig. 5d, from eq. 8) of these multistage cascades of differing network topologies. In α 2 S ()ω = . Fig. 5c, the signal output P3(ω) (from eq. 3), upon integra- − 2 (16) τω22+ tion by a three-tiered kinase cascade is shown. Taking ()d

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FigureIntegration 5 of differences in signal duration

Integration of differences in signal duration. Differences in signal duration parameterized by τdeg a.) time domain. b.) fre- − 1 2 quency domain; τ deg = 0.1 (dash-dotted), 0.5 (dashed), 1.0 (solid), 2.0 (dotted) lines. Plots of S(ω) ≡ |F(ω)| are shown. Corre- −1 −1 sponding plots of f(t) are shown in the inset. Short τ deg = 2.0 and long τ deg = 0.5 duration signals are filtered through b.) f f −1 −1 g3(ω) and c.) g1 ()ω resulting in b.) P3(ω) and c.) P1 ()ω for τ deg = 0.5 (dashed lines) and τ deg = 2.0 (dotted lines). Parame- + + − − f ters taken to be: b.) c) k1 = 2.0, k2 = 1.0, k1 = 1.0, k2 = 0.01, k = -5.0.

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+− Another network consisted of a sequence of steps in the kk==10. for i ∈ 1, 2, 3, g (ω) effectively filters the ii 3 form of biochemical intermediates in which the output is −1 short (τ d = 2.0) duration signal and results in an output connected to a downstream feedback loop or an interact- ing product. The gain in this network can have non P (ω) of small magnitude at all time scales 2πω-1 in the 3 monotonic behavior in which the low frequency compo- frequency spectrum. In contrast, for the signal character- nents of the signal are also filtered at time scales commen- −1 surate with the induction of the regulatory loop. This ized by τ d = 0.5, signal processing through g3(ω) results behavior enables the network to filter out signals of long in a signal of larger amplitude. The ratio of amplitudes duration. The minimal topological features of these bio- (with the superscript denoting the duration used) chemical networks provide distinct and robust mecha- τ =05. P d nisms for integrating signals that persist with different 3 at the optimal frequency (ω = 0) for the two sig- τ =20. characteristic time scales. As different temporally regu- P d 3 lated signals often lead to different transcriptional pro- τ =05. P d grams such as in NF-κB signaling [26,27], it is tempting to nals is 3 ≈ 17. speculate on the role that such filtering mechanisms may τ d =20. P3 have in regulating gene expression.

In Fig. 5d, the signal output P f ()ω (from eq. 6), obtained Acknowledgements 1 This work was funded by an NIH Director's Pioneer Award and NIH PO1 from a signal output that is also affected by a downstream AI071195-01 awarded to Arup Chakraborty. I thank Arup Chakraborty and negative (υ = -1) feedback loop, is shown. Parameters Kevin Fowler for helpful comments pertaining to this work. + + − − f used are: k1 = 2.0, k2 = 1.0, k1 = 1.0, k2 = 0.01, k = - References −1 1. Chen LF, Fischle W, Verdin E, Greene WC: Duration of nuclear 5.0. For the signal of long duration τ d = 0.5, only the NF-kappa B action regulated by reversible acetylation. Sci- ence 2001, 293:1653-1657. small frequency components of the signal are integrated. 2. Chen YR, Wang XP, Templeton D, Davis RJ, Tan TH: The role of c- This behavior is in contrast with the signal output of a Jun N-terminal kinase (JNK) in apoptosis induced by ultravi- olet C and gamma radiation – Duration of JNK activation −1 short duration signal τ d = 2.0. The amplitude difference may determine cell death and proliferation. Journal of Biological Chemistry 1996, 271:31929-31936. f ,.τ =05 P d 3. Dolmetsch RE, Lewis RS, Goodnow CC, Healy JI: Differential acti- in this case is 1 ≈ 0.2. vation of transcription factors induced by Ca2+ response f ,.τ =20 P d amplitude and duration. Nature 1997, 386:855-858. 1 4. Murphy LO, Blenis J: MAPK signal specificity: the right place at the right time. Trends in Biochemical Sciences 2006, 31:268-275. Discussion 5. Murphy LO, Smith S, Chen RH, Fingar DC, Blenis J: Molecular inter- pretation of ERK signal duration by immediate early gene Our models illustrate features of biochemical pathways products. Nature Cell Biology 2002, 4:556-564. that allow for the discrimination of signals that only differ 6. Santos SDM, Verveer PJ, Bastiaens PIH: Growth factor-induced in their duration. It is important to note that many impor- MAPK network topology shapes Erk response determining PC-12 cell fate. Nature Cell Biology 2007, 9:324-U139. tant, nonlinear effects, at the expense of analytical tracta- 7. Sasagawa S, Ozaki Y, Fujita K, Kuroda S: Prediction and validation bility, were excised in making the linear, weakly activated of the distinct dynamics of transient and sustained ERK acti- vation. Nature Cell Biology 2005, 7:365-U331. cascade approximation. For example, nonlinear positive 8. Marshall CJ: Specificity of Receptor Tyrosine Kinase Signaling feedback is known to give rise to bistability. Also nonlin- – Transient Versus Sustained Extracellular Signal-Regulated ear negative feedback can lead to oscillatory behavior. Kinase Activation. Cell 1995, 80:179-185. 9. Asthagiri AR, Lauffenburger DA: A computational study of feed- These effects, however, correspond to long time, steady back effects on signal dynamics in a mitogen-activated pro- state behavior and the present analysis focused on tran- tein kinase (MAPK) pathway model. Biotechnology Progress 2001, 17:227-239. sient signals of different duration and it is therefore 10. Ferrell JE, Machleder EM: The biochemical basis of an all-or- expected that such nonlinear effects are not expected to none cell fate switch in Xenopus oocytes. Science 1998, influence the qualitative behavior of the results in this 280:895-898. 11. Swain PS, Siggia ED: The role of proofreading in signal transduc- study. tion specificity. Biophysical Journal 2002, 82:2928-2933. 12. Thattai M, van Oudenaarden A: Attenuation of noise in ultrasen- In summary, we computed the frequency dependent sitive signaling cascades. Biophysical Journal 2002, 82:2943-2950. 13. Chaves M, Sontag EA, Dinerstein RJ: Optimal length and signal internal gain for two classes of biochemical pathways amplification in weakly activated signal transduction cas- involving multiple stages of regulation. The first model cades. Journal of Physical Chemistry B 2004, 108:15311-15320. 14. Heinrich R, Neel BG, Rapoport TA: Mathematical models of pro- consisted of a cascade of steps and showed how changes tein kinase signal transduction. Molecular Cell 2002, 9:957-970. in the number of steps as well as the amplification/atten- 15. Locasale JW, Shaw AS, Chakraborty AK: Scaffold proteins confer tion of the signal changed the networks' ability to filter diverse regulatory properties to protein kinase cascades. Proc Natl Acad Sci USA 2007, 104:13307-13312. high frequency (short duration) components of a signal.

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