Signal Duration and the Time Scale Dependence of Signal Integration in Biochemical Pathways
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Signal Duration and the Time Scale Dependence of Signal Integration in Biochemical Pathways The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters 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 Page 1 of 11 (page number not for citation purposes) BMC Systems Biology 2008, 2:108 http://www.biomedcentral.com/1752-0509/2/108 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. Page 2 of 11 (page number not for citation purposes) BMC Systems Biology 2008, 2:108 http://www.biomedcentral.com/1752-0509/2/108 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].