Optimality in kinetic proofreading and early T-cell recognition: revisiting the speed, energy, accuracy trade-off
Abstract
In the immune system, T cells can quickly discriminate between foreign and self ligands with high accuracy. There is significant evidence T-cells achieve this remarkable performance utilizing a network architecture based on kinetic proofreading (KPR). KPR-based mechanisms actively consume energy to increase the specificity beyond what is possible in equilibrium. An important theoretical question that arises is to understand the trade-offs and fundamental limits on accuracy, speed, and dissipation (energy consumption). Recent theoretical work suggests that it should always be possible to reduce the error of KPR-based mechanisms by waiting longer and/or consuming more energy. Surprisingly, we find that this is not the case and that there actually exists an optimal point in the speed-energy-accuracy plane for KPR and its generalizations. We give general arguments for why we expect this optimal to be a generic property of all KPR-based biochemical networks and discuss implications for our understanding of the T cell receptor circuit.
pacs:
Valid PACS appear hereA central problem in immunology is the recognition of foreign ligands by the immune system. This process is carried out by specialized immune cells called T-cells which activate the immune response in the presence of foreign ligands. Foreign ligands are presented to T-cells by specialized Antigen Presenting Cells (APCs) that bind a repertoire of self and foreign peptides. As shown in Fig. 1, T-cells activation occurs when specialized receptors on the surface of T-cells, called T-cell receptors (TCRs), bind APCs, and activate downstream the TCR signaling network, leading to an immune response.
It has been shown that T-cells have a high sensitivity to foreign ligands. A few foreign ligands (less than 10) appearing on the membrane of a T-cell are able to trigger the immune responseIrvine et al. (2002); Chakraborty and Weiss (2014). Moreover, this decision is made extremely quickly: it only takes 1-5 mins to make the decision to activate or not Stoll et al. (2002). Despite the speed with which the response is mounted, T-cells can accurately sense the existence of foreign ligands with an error rates as small as Mckeithan (1995); Alon (2006). This raises natural questions about how the T-cell signaling network can operate with such high speed, sensitivity, and accuracy.
Experimental evidence suggests that T-cell activation is set by the binding time of the antigen-receptor complex Feinerman et al. (2008); François and Altan-Bonnet (2016). If the binding time of the ligand to the receptor is below a sharp threshold (3-5 sec), T-cells do not activate. However if the binding time is above this threshold, T-cells activate with extreme sensitivity. This so called ‘life-time’ dogma places stringent conditions on the machinery of the immune responseFeinerman et al. (2008). A lot is known about the biochemical networks that implement this thresholding procedure. The receptor-ligand complexes go through multiple rounds of phosphorylation (throughout we denote the number of phosphorylations by ). Within the life-time dogma, an immune response is triggered if the concentration of the ligand-receptor complex that has been phosphorylated times exceeds a threshold concentration.
The ability of T-cells to discriminate between foreign and self ligands arises from the difference in the binding times of foreign () and self () ligandsGascoigne et al. (2001). Typically, in the immune system, and . In equilibrium, this binding time difference cannot account for the incredible accuracy of the T-cell immune response. Detailed balance places constraints on the chemical reaction rates and the reliability of the discrimination process is ultimately limited by equilibrium thermodynamicsSartori and Pigolotti (2015). This binding time difference can be directly translated into a difference in binding free energies of foreign and self ligands Hopfield (1974); Ninio (1975). Thus, a biochemical network that works at equilibrium can achieve a minimum error rate of , nearly three orders of magnitude smaller than that seen in experiments.
It is known the immune system can beat this bound by working out-of-equilibrium and consuming energyMckeithan (1995). It is now thought that the T-cell employs a form of kinetic proofreading(KPR), first proposed by HopfieldHopfield (1974) and NinioNinio (1975). But current understanding of KPR and its implications for immune response have several weaknesses: firstly, most theoretical treatments of KPR involve approximating certain reactions as irreversible making it difficult to consistently calculate energy consumption; second, it is extremely hard for KPR-based schemes to simultaneously distinguish ligands with similar binding times and operate over a large dynamic range of ligand concentrations.The later shortcoming has been addressed by a generalization of KPR called “adaptive sorting”. In adaptive sorting, an additional feedback couples the KPR cascades in the T-cell through a common kinase that regulates all the phosphorylation of all T-cell receptors Lever et al. (2014); François and Siggia (2008); François and Altan-Bonnet (2016); Lalanne and François (2013); François et al. (2013).
A fundamental issue in the study of T-cell activation is to understand the trade-off between different functionalities – accuracy, speed and dissipation – in the immune discrimination process. Many works have studied the relation between accuracy and dissipation or accuracy and speed for some KPR-based biochemical networkSavageau and Freter (1979); Ehrenberg and Blomberg (1980); Freter and Savageau (1980); Qian (2006); Murugan et al. (2012); Rao and Peliti (2015); Mehta et al. (2016); Banerjee et al. (2017); Das (2016). Some others have discussed general error rate bounds under power constraints in the context of thermodynamics or information theoryLandauer (1961); Bialek and Setayeshgar (2005); Mora (2015); Lang et al. (2014); Laughlin (2001); Qian (2003). Few worksLan et al. (2012); Lahiri et al. (2016) consider the trade-off between these three quantities simultaneously. These theoretical work suggests that it is always possible to reduce the error of KPR-based mechanisms by waiting longer and/or consuming more energy Savageau and Freter (1979); Murugan et al. (2012).
In this paper, we calculate the speed, power dissipation, error rate and output signal (the combined concentration of and ) explicitly for a KPR-based biochemical network, with and without a feedback that implements adaptive sorting (shown in Fig. 2). We ask if there is an optimal operating point for T-cell activation networks where T cells can make fast and accurate decisions while utilizing energy efficiently. Surprisingly, we find that such an optimal point exists for KPR and its generalizations. Near the optimal point, the response time and power dissipation are consistent with those observed in experiments, implying that many mechanisms of early T-cell recognition are well described by KPR-based models.
I Model
We start from the daptive sorting model shown in Fig. 2 François and Siggia (2008); François et al. (2013); François and Altan-Bonnet (2016); Lalanne and François (2013). The receptor, , can bind a foreign or self ligand, to form a complex and respectively. This complex can be phosphorylated a maximum of times. We denote a receptor-ligand complex that has been phosphorlyated times by with for foreign ligands and for self ligands. The dynamics of the biochemical network can be written as:
(1) | |||||
where , , and . and are the free concentration of receptors and ligands, with , and the total number of receptors, ligands and kinase respectively. For notational simplicity, throughout the manuscript we assume that cell volume is fixed and hence do not distinguish between species number and concentration. In Fig. 2, we set and . More information about molecular species and notation can be found in Table. 1.
In the adaptive sorting network, both foreign and self ligands can bind a receptor and form the receptor-ligand complex, , which can undergo multiple rounds of phosphorlyation ( goes to ) and dephosphorylation ( goes to ). The receptor-ligand complexes can disassociate (at a rate for self ligands and for foreign ligands). During this process, the phosphate groups are lost and and whole process reinitiates. Importantly, once a ligand is bound to a receptor, it is impossible for the biochemical machinery to distinguish between foreign and self ligands. The binding rate , the phosphorylation rate, , and the dephosphorylation rate, , inside the cell are the same for the foreign and self ligands and the only difference between foreign and self ligands are the lifetimes of their corresponding receptor-ligand complexes. For this reason, the decision to activate is based on the concentration of the total final products from both the foreign ligand () or self ligand ().
In the adaptive sorting network, in addition to the phosphorylation cascade, a negative feedback is used to modulate the phosphorylation and/or dephosphorylation rates François and Siggia (2008); François and Altan-Bonnet (2016). For example, in Fig. 2 the last phosphorylation step, from to , is modulated by the level of active kinase , which itself is dependent on the concentration of the -th intermediate concentration through phosphorylation. With this feedback, the output signal is independent of the ligand concentration and only replies on the value of . This model reduces to a KPR cascade when the feedback is absent, and .
In most treatments of KPR, the disassociate of the receptor-ligand is often treated as an irreversible process (). Often, this is a good approximation since phosphatases can easily bind free receptors and quickly remove phosphate groups from the receptors Mckeithan (1995). However, in any thermodynamically consistent model, all reactions are reversible and it is important to consistently treat both the forward rate and backward rate for the formation and disassociation of a complex. For this reason, we introduce a small rate, , for directly forming a complex and (see Supporting Information). This functional form is a direct consequence of assuming that there is a constant free energy difference per phosphorylation. Furthermore, we assume this rate is small (). Below, we show that taking is essential to obtain the optimal point in the speed-energy-accuracy plane.
Symbol | Definition |
---|---|
Agonist complex phosphorylated n times | |
Non-agonist complex phosphorylated n times | |
Receptor | |
Active kinase | |
Inactive kinase | |
Kinase | |
Complex phosphorylation rate | |
Complex phosphorylation rate at the final step | |
Complex dephosphorylation rate | |
Kinase phosphorylation rate | |
Kinase dephosphorylation rate |
Ii Defining Accuracy, Speed, and Dissipation
Before analyzing the biochemical network outlined above, it is necessary to define accuracy, energy consumption, and speed for T-cell recognition in greater detail.
ii.1 Accuracy
Recall, that a T-cell makes the decision to activate based on the total concentration of the full phosphorylated complexes from both the foreign ligand () and self ligand (). Ideally, T-cells are activated only in response to foreign ligands. Thus, following Hopfield Hopfield (1974) we can define the error rate as the ratio of and :
(2) |
The concentrations of different components can be calculated by solving the deterministic equations (I) at steady state. In the immune recognition by T cells, it is important to achieve a small error rate . For an irreversible -step KPR process (i.e. ), can reach a minimum value we dub the “Hopfield limit”
(3) |
We define the accuracy as one minus the error rate, .
ii.2 Energy Consumption
In any non-equilibrium steady state, detailed balance is broken and leading to the existence of net currents in the networkLandauer (1961); Hill (2012). The chemical potential difference between the reactants and products can be written as
(4) |
where are forward- and backward-reaction fluxes. The net current is . The power dissipation is defined as Hill (2012); Qian (2007)
(5) |
For example, the power dissipation of the first-step phosphorylation process: \ce ¡=¿[\ce][\ce] can be calculated as:
(6) |
This can be generalized to the full KPR cascade and adaptive network (see Supporting Information). Finally, we adapt the convention of nonequilibrium thermodynamics and use the phrases “energy consumption” and “power dissipation” interchangeably.
ii.3 Speed
The speed of decision-making process is related to the mean first passage time(MFPT) of a stochastic processSrivastava and Leonard (2015). The MFPT is defined as average the time taken to produce one molecule of the final product from the foreign ligand . For example, at each time step, one molecule of the complex can be phosphorylated at a rate to yield , or can be dephosphorylated at a rate to get a molecule to , or alternatively decay rate to yield a free receptor . Microscopically, this can be viewed a stochastic process – similar to a random walk– and different realization of this process will take different amounts of time. The MFPT is taken as the average time it takes to complete to get from the starting point to the target. We use the mean MFPT to define the inverse of the decision speed. Detailed calculation procedures can be found in Supporting Information and Bel et al. (2009).
Calculating speed in the adaptive sorting network is technically much more challenging than in KPR due to the non-linearity introduced by the additional feedback loop. To overcome this difficulty, we employ a linear-response approximation around the steady-state operating point when calculating the speed. Such linear-response approximations are commonly employed in engineering (e.g. gain, bandwidth) and have been adapted with great success to analyze biochemical circuits Detwiler et al. (2000). In the linear-response regime of adaptive sorting, the MFPT can be calculated using methods analogous to KPR (see Supporting Information and Van Kampen (1992); Mehta and Schwab (2012)).
Finally, we note that in a related work, the speed is associated with the inverse of the smallest eigenvalue of the master equation describing the biochemical circuitLahiri et al. (2016). However, it has been shown that this definition is not a good measure of speed unless one considers long Markov chains (i.e. ) dominated by nearest-neighbor transitions Kim (1958). The circuits considered here operate very far from these regimes.
Iii Results
We now analyze the speed-energy-accuracy tradeoff in KPR and adaptive-sorting circuits. One difficulty involved in identifying general principles are the large number of parameters whose choice can dramatically change the properties of the underlying circuit (see Table. 1). For this reason, we will take a strategy based on randomly sampling these parameters in numerical simulations and looking for accessible regions in the energy-speed-accuracy plane. This spirit is similar to the one used to identify robustness in the adaptation circuit of bacterial chemotaxis Barkai and Leibler (1997); Ma et al. (2009). We begin by analyzing a KPR cascade where the feedback loop from the kinase in Fig. 2 is turned off and then subsequently extend our analysis to the full adaptive sorting network.
iii.1 Kinetic proofreading
Recent theoretical work suggests that it is always possible to reduce the error of KPR-based mechanisms by waiting longer and/or consuming more energyMurugan et al. (2012). Surprisingly, we find that this is not the case. Our results show the error rate increases dramatically at extremely slow speeds/low dissipation when (the rate to directly form a complex) has a nonzero value.
We studied the effects of varying with numerical simulations shown in Fig. 3. When , waiting longer always decreases the error rate. As shown in Fig. 3(a), the error rate monotonically decreases the with the MFPT (1/speed) and asymptotically reaches the Hopfield limit for an infinitely slow circuit: for a circuit with phosphorylations. In this high accuracy regime, a ligand must bind the receptor multiple times and transverse all steps of the phosphorylation cascade before reaching the final products . However, when , for sufficiently long times, the probability to directly form a phosphorylated complex and bypass the initial kinetic proofreading steps becomes non-negligible. This leads to an increase in the error rate Hopfield (1974); Murugan et al. (2012). Thus, increasing drives a cross-over in the dynamic behavior of the biochemical circuit from a regime where waiting longer increases the accuracy to one where waiting longer decreases the accuracy. We also investigated the relationship between the speed of the circuit and power consumptions. Fig. 3(b) shows that over large parameter regime, the energy consumption and MFPT (1/speed) exhibit an approximate power law (linear relationship on a log-log plot). This indicates that making a decision quickly always requires a a large amount of energy consumption. This approximate power-law relationship breaks down for extremely slow circuits.
In order to better understand the relationship between speed, accuracy, and energy consumption, we randomly sampled different combinations of the three parameters: , , and calculated all three quantities(see Supporting Information for details). The results are shown in the Fig. 4(a). We also calculated the total output signal (the concentration of ) for each parameter set Fig. 4(b). In both plots, each point corresponds to a different choice of the parameters.
To better understand these plots, it is helpful to separate the parameters into four qualitatively distinct operating regimes (see Fig. 4): (A) a high-accuracy regime, (B) a high-speed, low-dissipation, low-accuracy regime, (C) a high-dissipation, low-accuracy regime, and (D) a low-dissipation, low-speed, low-accuracy regime.
One of the most dramatic features in Fig. 4(a) is the blue, high-accuracy region A. In Region A, the error rate of the KPR cascade approaches its theoretically minimum possible value (i.e. the “Hopfield Limit”) . This high accuracy region is realized when , and . These parameter regimes corresponds to the assumptions outlined by Hopfield as being necessary for achieving high-accuracy proofreading Hopfield (1974). Many choices of parameters in Region A achieve this high accuracy. However, as shown Fig. 4(b) for many of these choices of parameters the magnitude of the output signal is quite small. This motivates defining an optimal operating point of the KPR regime as the choice of parameters with highest accuracy and a high output signal. This point is marked as the optimal point in Fig. 4(a). We discuss this optimal point extensively below.
In Region B, one can make a fast decision speed with minimal energy consumption, but the error rate is well above the Hopfield limit. Here, and . In this parameter regime, there is a steady- flux of empty receptors that are converted to the fully phosphorylated output complex. The MFPT is reduced but the system becomes insensitive to the difference between foreign and self-ligand binding times: the forward rate is so large that there is no time for the intermediate complexes to decay making it impossible to distinguish and .Region C has the highest error rate. Here, , and . For such large values of , there is a continuous flux from free receptor directly to the fully-phosphorylated complex , with most output molecules bypassing the proofreading steps. In this region, is much bigger than the binding times of ligands resulting in error rates that can be as large as (see Supporting Information). In practice, for reasonable values of (e.g. ), no biochemical networks operate in region C. Finally, in region D, speed decreases dramatically because of , .
Figure 4c and d show cross-sections of the error rate for a fixed speed and fixed dissipation rate respectively. These graphs were generated by selecting all parameters that lie along the vertical and horizontal dashed lines in Fig. 4a. One of the most striking aspects of these plots is how dramatically the error rate decreases from the “equilibrium value” of to the theoretical maximum “Hopfield limit” as a function of the dissipation rate and mean first-passage time. A similar plot for speed versus error rate in a recently obtained by Banerjee et al. (2017). Furthermore, the transition between these values become steeper and narrower as is reduced. These plots suggest that for slow speeds (above ) and low dissipation rates (below ) there is likely a dynamic phase transition in the KPR circuit when either the dissipation rate or speed is held fixed and other parameters are varied.
Murugan and collaborators have argued that KPR has a natural mapping to microtubule growth, a system with a known dynamical phase transition between growth and shrinkage, and it has been argued that such a transition is also likely to be a generic feature of KPR Murugan et al. (2012). However, unlike the systems analyzed by Murugan et al. (2012), we consider a non-zero transition rate, , which leads to qualitatively different results. In particular, when the mean first-passage time become comparable to the typical time it takes to “bypass” the KPR steps and directly form the complex , set by (see Fig. 2). This leads to the low-fidelity region C in Fig. 4 and accounts for the existence of the optimal point in the speed-accuracy-dissipation plane.
iii.2 Extending our results to adaptive sorting
In the preceding section, we have focused on the speed, accuracy, and dissipation trade-offs in a simple KPR cascade. Adaptive sorting is a very promising extension of KPR relevant for understanding T-cell activation in immune recognition François and Siggia (2008); François and Altan-Bonnet (2016); Lalanne and François (2013); François et al. (2013). Adaptive sorting employs an additional negative feedback loop in the last step of the KPR cascade that ensures the output signal is independent of the number of ligands in the environment. This ability to perform “absolute ligand discrimination” is a key feature of adaptive sorting. It accounts for how a T-cell can achieve high accuracy in natural environmental conditions where the concentration of self-ligands is large and dwarfs the concentration of foreign ligands ( and ). A natural question is to ask if there is any tradeoffs involved needed to achieve absolute ligand discrimination. One such tradeoff is antagonism, where increasing the concentration of foreign ligands actually degrades the response of the adaptive sorting circuit Francois et al. (2016). We show here that there is another tradeoff between absolute ligand discrimination and the speed at which the T-cell receptor circuit can operate.
Figure 5 shows error rate, mean first-passage time, and dissipation rate of the adaptive sorting and the KPR cascade analyzed above with regards to the tradeoffs between speed-accuracy and dissipations . The dissipation and error rate of the adaptive sorting model is comparable to a KPR cascade. However, from Fig. 5(a,b), it takes the adaptive sorting circuit much longer to achieve a similar error rate as a KPR. For a very large input signal, the phosphorylation rate of the last step in the cascade is dramatically decreased, leading to dramatic decrease in speed because most complexes fall apart before reaching the final step of the cascade. Furthermore, notice that unlike KPR, the adaptive sorting circuit is unable to achieve even modest error rates for mean first passage times of 100s (vertical dashed lines in Fig. 5), corresponding to the experimentally observed time it takes T-cells to make the activation decision.
Iv Discussion
The immune system must quickly and accurately recognize foreign ligands. To carry out this task, the T-cells work out of equilibrium by actively consuming energy. This raises natural questions about the relationship between speed, accuracy, and energy consumption in two classes of biochemical networks that have been used to model immune recognition: a KPR-based network and a generalization of KPR, adaptive sorting. Importantly, unlike previous work, we made no approximation about the underlying parameter space and this allowed us to identify an optimal operating point in the speed-energy-accuracy plane. We also found that the behavior of these networks exhibit four different regimes, which surprisingly included a fast, high-accuracy regime at intermediate energy consumption.
Our results stand in contrast with recent theoretical work suggesting that it may always be possible to achieve a better accuracy by waiting longer or consuming more energy. The underlying reason for this is that unlike these previous works we allow for a tiny (but) non-zero rate for bypassing the proofreading steps. This non-zero rate is necessary in any thermodynamically consistent model. While the parameter has no effect at short times, for very long times the error increases because the probability of bypassing the proofreading steps becomes significant. The generality of this argument suggests that our conclusions should also hold for other, more complicated biochemical networks.
It has been argued that a KPR-based T-cell activation is likely to fail when the concentration of external ligands becomes large and one must instead consider an adaptive sorting based circuit François and Siggia (2008); François and Altan-Bonnet (2016); Lalanne and François (2013); François et al. (2013). Unlike a simple KPR cascade, the adaptive sorting network can distinguish between foreign and self even for large ligand concentration, a property dubbed “absolute ligand discrimination”. We have found that absolute ligand discrimination comes at a large cost in speed compared to a simple KPR-based circuit.
We can compare our results for speed accuracy, and energy consumption to experiments. T-cells spend 1-5 mins to make the decision to activate Feinerman et al. (2008). A rough estimation of the error rate from experiment suggests cells can achieve error rates in the range or smaller, with the exact number depending on properties of ligands Mckeithan (1995); Alon (2006). The energy expended by a T-cell to make the activation decision is hard to measure directly. However, estimates of the power consumption from glucose consumption suggest a typical cells uses about Pollard and Borisy (2003); Milo et al. (2010). These numbers set strict experimentally-derived bounds for our model.
For a circuit with phosphorylations, the minimum error rate achieved by both KPR and adaptive sorting is , on par with the experimental error rates. As shown in Fig. 5, the KPR cascade can achieve close to this optical accuracy in the experimentally observed decision time of . The power consumption of the circuit is (where we have used the standard conversion Rosing and Slater (1972)), just one-one thousandth of the total energy budget of the cell. Moreover as shown in Fig. SI2(see Supporting Information), increasing the number of steps in the phosphorylation cascade can significantly increase the accuracy of a KPR cascade with only modest decreases in the speed and the magnitude of the output signal. An adaptive sorting circuit can also reach the optimal error rate of using approximately the same energy budget as a simple KPR cascade. However, the absolute ligand discrimination of adaptive sorting comes at a steep price in terms of speed. For the biologically realistic window for making immune recognition, the KPR cascade achieves a respectable error rate between and whereas the adaptive sorting circuit is essentially non-functional.
More generally, the trade-off between speed, accuracy, and power consumption in realistic biochemical networks is still poorly understood. Our results based on a simple model of immune decisions show that thermodynamics places strict constraints on these non-equilibrium processes. Energy consumption is required to maintain these non-equilibrium processes. With extremely low energy consumption or slow speed, the decision signal will be ruined by thermal fluctuations. However, when operating in regimes with extremely large energy consumption or speed, subtle effects can suddenly transition circuits so that decisions are dominated by rare events that destroy accuracy. This suggests that great care is needed in both modeling and/or engineering KPR-based decision making circuits.
One of the most striking aspects of our simulations are the sudden transitions in accuracy as a function of the dissipation rate (at fixed speed) or speed (at fixed dissipation). This transitions seem to be indicative of an out-of-equilibrium dynamic phase transition. In the future, it will be interesting to further investigate this transition and see if it is possible to adopt analytic methods and fluctuation-type theorems to better understand its origins. Our work also suggests that it is extremely difficult for adaptive sorting networks to simultaneously perform absolute ligand discrimination and operate quickly. An important area of future work is to better understand if this trade-off is fundamental or can be bypassed with more clever network architectures. Finally, it will be interesting to explore general networks and develop analytic techniques to further our understanding optimal operating points with regards to speed, accuracy, and power consumption.
Acknowledgements.
This work was supported by NIH NIGMS MIRA grant number 1R35GM119461 and Simons grant in the Mathematical Modeling of Living Systems to PM.Appendix A Definition of Model and parameter choices
A schematic of the model we are considering is shown in Fig. 6. As described in the main text, we denote a receptor-ligand complex that has been phosphorlyated times by with for foreign ligands and for self ligands. Furthermore, we denote the maximum number of phosphorylations as N. With this notation, using the law of mass action, we have
(7) | |||||
where , , and . Typically, we set: , , , , , , , , and . Any deviations from this choice of parameter is explicitly noted.
a.1 Accuracy
At steady state, the error rate can be written as
(8) |
In the presence of the kinase feedback , the set of eqs. (A) are no longer linear and but the steady-state solution can still be found easily using an iterative method.
a.2 Energy Consumption
The power dissipation is calculated based on the net flux and the chemical potential differenceHill (2012); Qian (2007). We define the net flux , at in the main pathway.
Considering the flux conservation, the power dissipation can be written as
The total power dissipation is from the contribution of both foreign and self ligands: .
a.3 Role of
In KPR, the reversible decay rate is ignored as it has extremely small value. However, from thermodynamics and calculation for energy consumption, we have to include it into our model. One choice of the reversible decay rate is . There are two reasons for this form: 1. the production rate from ligands and receptors to should be smaller then the one to as one more phosphorlyation step is involved; 2. it is also natural to assume the energy consumption is the same for each phosphorylation step .
The free energy difference between n and n+1 phsphorlyation round can be calculated as:
a.4 Speed
The speed is defined by the mean first passage time(MFPT) for the foreign ligand. Here we mainly follow the procedures in Ref. Bel et al. (2009). The concentration vector is defined as . An final ’dark’ state is added because the response is only activated at the end and it can be treated as absorbing markov chain. Added this absorb state, it becomes an irreversible process, which is impossible to calculate the energy consumption. The transfer probability from to the ’dark’ state is (irreversible). We set , a large value, which means the final step has little effect on MFPT. Without loss of generality, we begin with and , which can be generalized other cases easily. The master equations eqs. (A) can be rewritten as and
(10) |
But eqs. (A) are not linear. The first order perturbation approximation is adapted and we can linearize (with bar denoting average) to get .
(11) |
where is
(12) |
Applying the Laplace transform, , the master equations can be rewritten as:
(13) |
The MFPT can be written:
(14) |
which can be calculated numerically. It should be notified that the concentration and probability have the same master equations but a different pre-factor. When choosing the initial condition , the pre-factor is set to be 1 and solved from eq. (13) is exactly a probability distribution .
Appendix B Simulation Details for Phase Diagram
In this figure, we run samples with random sets log uniformly choosen between , , .
It can be observed that a large amount of red points distributes over regimes C and D with . This is because of and the inverse flux at the final step dominates. In the extreme case: is very large, will occupy most of products and free ligands have little concentration.
As dominates,
Appendix C Changing the number of phosphorylation steps
Here, we show simulations for the KPR-cascade when we vary the maximum number of phosphorylation steps .
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