Energy dissipation in an adaptive molecular circuit
EEnergy dissipation in an adaptive molecular circuit
Shou-Wen Wang , , Yueheng Lan and Lei-Han Tang , Department of Engineering Physics, Tsinghua University, Beijing, 100086, China Beijing Computational Science Research Center, Beijing, 100094, China Department of Physics, Tsinghua University, Beijing, 100086, China Department of Physics and Institute of Computational and Theoretical Studies,Hong Kong Baptist University, Hong Kong, China
Abstract.
The ability to monitor nutrient and other environmental conditions with highsensitivity is crucial for cell growth and survival. Sensory adaptation allows a cellto recover its sensitivity after a transient response to a shift in the strength ofextracellular stimulus. The working principles of adaptation have been establishedpreviously based on rate equations which do not consider fluctuations in a thermalenvironment. Recently, G. Lan et al. (Nature Phys., 8:422-8, 2012) performed adetailed analysis of a stochastic model for the E. coli sensory network. They showedthat accurate adaptation is possible only when the system operates in a nonequilibriumsteady-state (NESS). They further proposed an energy-speed-accuracy (ESA) trade-off relation. We present here analytic results on the NESS of the model through amapping to a one-dimensional birth-death process. An exact expression for the entropyproduction rate is also derived. Based on these results, we are able to discuss the ESArelation in a more general setting. Our study suggests that the adaptation error canbe reduced exponentially as the methylation range increases. Finally, we show that anonequilibrium phase transition exists in the infinite methylation range limit, despitethe fact that the model contains only two discrete variables.
1. Introduction
As a paradigmatic example of environmental monitoring in biology, the E. colichemotactic sensory system has been studied extensively over the years [1]. Its corecomponent is the transmembrane methyl-accepting chemotaxis protein (MCP) receptor.MCP binds selectively to ligands outside the cytoplasmic membrane and modulatesthe activity of its downstream signal transduction pathway in a way that depends onits methylation state. Two aspects are recognized to be crucial to the performanceof the sensory network in the biological context: sensitivity of detection in a noisyenvironment, and adaptation to maintain that sensitivity over a broad range of ligandconcentrations. With regard to high sensitivity to diffusing chemicals in the surroundingmedium, Berg and Purcell [2] presented an optimal strategy in 1977 based on simplephysical considerations. They showed that the measured chemotactic sensitivity ofE. coli approaches that of the optimal design. Further indication of the organism’s a r X i v : . [ q - b i o . S C ] M a y nergy dissipation in an adaptive molecular circuit nergy dissipation in an adaptive molecular circuit
2. A model for sensory adaptation
Here, we briefly introduce the transmembrane methyl-accepting chemotaxis protein(MCP) receptor, which is the core component for receiving signal and exercisingadaptation [1]. MCPs regulate the clockwise-counterclockwise rotational switch ofdownstream flagellar motors which drive the run-and-tumble motion of an E. coli cell. Asimple cartoon of this receptor is illustrated in Figure 1(a). The activity of the receptorcan be described by a binary variable a : a = 1 for the active state and a = 0 forthe inactive state. The transition rate between the two states depends on the externalligand concentration (i.e., signal strength) s and the internal methylation level m . m ranges from 0 to m , with m = 4 for a single MCP. The methylation level can beincreased by enzyme CheR and decreased by enzyme CheB, in a way that depends onthe activity of MCP. Figure 1(b) illustrates response of the MCP receptor to a stepwisesignal s obtained from experimental measurements. The mean activity (cid:104) a ( t ) (cid:105) changessharply in a short time window τ a less than a second, and recovers slowly over a muchlonger time scale τ m , of the order of a minute, due to the slow change of average internalmethylation level (cid:104) m ( t ) (cid:105) . The output recovery after a transient response to externalstimuli is called adaptation. The performance of adaptation is characterized by theadaptation error which can be defined as the ratio between the final shift of activity andthe relative change of signal strength, as illustrated in Figure 1(b). (a) (b) Figure 1. (a) The MCP receptor. (b) Mean response of the MCP receptor to astepwise signal.
A Markov network model with internal states specified by ( a, m ) was proposed fora single MCP by Lan et al. [19], as shown in Figure 2. Transition between active and nergy dissipation in an adaptive molecular circuit τ a , with ratesgiven by ω ( m, s ) = 1 τ a exp (cid:16) β E ( m, s ) (cid:17) , ω ( m, s ) = 1 τ a exp (cid:16) − β E ( m, s ) (cid:17) , where β = 1 /T is the inverse temperature, and∆ E ( m, s ) = e ( m − m ) + f ( s )the free energy difference between states (0 , m ) and (1 , m ), with f ( s ) = ln[(1+ s/K i ) / (1+ s/K a )]. Here e > m an offset methylation level [20, 1],and K i and K a ( (cid:29) K i ) are equilibrium constants for ligand binding to the receptor in theinactive and active states, respectively. Transition between different methylation levelstakes place on the time scale τ m , with rates indicated in Figure 2: when the receptor isinactive, the rate of methylation (assisted by enzyme CheR) is K CR while the rate ofdemethylation is αK CR ; when the receptor is active, the rate of demethylation (assistedby CheB) is K CB while the rate of methylation is αK CB . An estimate of the typicalmethylation/demethylation cycle time is given by τ m = k − CB + k − CR .The parameter α specifies the degree of disequilibrium in the system. When α = α EQ ≡ exp( βe / α < α EQ ,the system is driven out of equilibrium with generally different properties which westudy using both analytical and numerical methods. Therefore α describes the strengthof driving to keep the receptor to operate under out of equilibrium conditions.In the numerical examples presented below, we adopt the parameter values assuggested in Ref. [19]: m = 1, K i = 18 . µ M, K a = 3000 µ M, β = 1 ( kT asthe unit of energy) and e = 2. The time constants are chosen as τ a = 0 . s , and k CB = k CR = 0 . s − . For this parameter set, α EQ = e . To simplify the notation, wewrite ω ( m, s ) , ω ( m, s ) as ω ( m ) and ω ( m ) respectively. Figure 2.
The Markov network model of a single receptor MCP in E. coli. Red arrowsindicate existence of a futile cycle at small α , which is essential for adaptation. nergy dissipation in an adaptive molecular circuit
3. Adaptation and the NESS
We first revisit the condition for adaptation first obtained by Lan et al. [19]. For themodel introduced in Sec. 2, the master equation for the joint probability P ( a, m ) takesthe form, dP (0 , m ) dt = k CR P (0 , m −
1) + αk CR P (0 , m + 1) + ω ( m ) P (1 , m ) − [ k CR + αk CR + ω ( m )] P (0 , m ) , (1 a ) dP (1 , m ) dt = αk CB P (1 , m −
1) + k CB P (1 , m + 1) + ω ( m ) P (0 , m ) − [ αk CB + k CB + ω ( m )] P (1 , m ) . (1 b )From the above, we obtain the evolution equations for the moments (cid:104) m (cid:105) = (cid:80) a,m mP ( a, m ) and (cid:104) a (cid:105) = (cid:80) m P (1 , m ): d (cid:104) a (cid:105) dt = (cid:88) m (cid:104) ω ( m ) P (0 , m ) − ω ( m ) P (1 , m ) (cid:105) , (2 a ) d (cid:104) m (cid:105) dt = (1 − α )( k CR + k CB ) (cid:16) −(cid:104) a (cid:105) + k CR k CR + k CB (cid:17) + B . (2 b )Here B = αk CR P (0 ,
0) + k CB P (1 , − k CR P (0 , m ) − αk CB P (1 , m ) depends on theprobabilities for the extreme methylation states m = 0 and m = m .In a steady environment of constant ligand concentration s , the system is expectedto reach a steady state in a time τ m where both (cid:104) m (cid:105) and (cid:104) a (cid:105) assume constant values.Setting the right-hand-side of Eq. (2 b ) to zero yields, (cid:104) a (cid:105) = a s = k CR k CR + k CB + 11 − α B k CR + k CB . (3)Since the methylation and demethylation rates k CR and k CB are assumed to be constantsin the model, the first term a ≡ k CR / ( k CR + k CB ) on the right-hand-side of Eq. (3) isindependent of s . Figure 3 shows a s against s for three different values of α , obtainedfrom numerically exact solution of the model in the NESS. The steady-state activity a s is centered around a (dashed line) over a large range of s for α <
1, but not so for α ≥
1. The “adaptation error” (cid:15) ≡ | a s − a | = (cid:12)(cid:12)(cid:12) − α B k CR + k CB (cid:12)(cid:12)(cid:12) (4)is essentially controlled by the size of the boundary term B . For α < B is small overa broad range of s . As we shall see in the next section, the NESS distribution in thiscase is indeed centered in the middle of the allowed methylation range. This is howevernot the case when α > α < α >
1. Figure 4 shows our results obtained by numerically integratingthe master equations (1 a ) and (1 b ) at three different values of α , upon a jump in ligandconcentration from 10 K i to 15 K i at t = 0. The initial response of (cid:104) a (cid:105) to the signal nergy dissipation in an adaptive molecular circuit s ignal: s/K i a s α = 0 . α = 1 α = α E Q
Figure 3.
Mean receptor activity against ligand concentration for three differentvalues of the nonequilibrium parameter α . The dash line indicates the value a . Herethe methylation range m = 4. ramp is qualitatively similar in the three cases, i.e., a fast depression of receptor activityto a near plateau value in a time of order τ a . However, opposite behavior is seen atlonger times, in concert with the change in methylation level as seen in Fig. 4(b). For α >
1, a further decrease of the mean activity is seen when the methylation level startsto decrease in response to the change in (cid:104) a (cid:105) . On the other hand, when α <
1, themethylation level increases in accordance with Eq. (2 b ), eventually restoring the meanactivity to a value close to the pre-stimulus level. The latter is precisely the scenario foradaptation that employs a change in the methylation level to offset the activity changeeffected by the shift in signal strength.In summary, both the steady-state activity and the transient response to a signalramp show qualitatively different behavior below and above α = 1. We thus concludethat the condition for adaptation in this model is α < In the previous subsection, we obtained the condition for adaptation by considering themoment equations with the help of numerical integration of the master equation. Togain a complete understanding of the NESS, it is necessary to calculate the distributionfunction P ( a, m ). Fortunately, for the model in question, this can be done under the“fast equilibrium” approximation facilitated by the separation of the time scales τ a and τ m (cid:29) τ a . Then, P (1 , m ) and P (0 , m ) satisfy the local detailed balance P (1 , m ) P (0 , m ) = ω ( m ) ω ( m ) + O ( τ a /τ m ) , (0 ≤ m ≤ m ) . (5)Let P ( m ) ≡ P (0 , m ) + P (1 , m ), we obtain, P (0 , m ) = 11 + exp[ − β ∆ E ( m, s )] P ( m ) , P (1 , m ) = exp[ − β ∆ E ( m, s )]1 + exp[ − β ∆ E ( m, s )] P ( m ) . (6) nergy dissipation in an adaptive molecular circuit −2 −0.06−0.05−0.04−0.03−0.02−0.010 t h a ( t ) − a ( ) i α = 0 . α = 1 α = α E Q (a) −2 −0.2−0.15−0.1−0.0500.050.10.150.2 t h m ( t ) − m ( ) i α = 0 . α = 1 α = α E Q (b)
Figure 4.
An initial steady state at s = 10 K i is perturbed by shifting the ligandconcentration to s = 15 K i at t = 0. Results at three different values of α are shown:(a) change in the mean receptor activity (cid:104) a ( t ) (cid:105) against t ; (b) change in the methylationlevel (cid:104) m ( t ) (cid:105) against t . The methylation range m is set to be four. With the help of (5), Eqs. (1 a ) and (1 b ) combine to yield dP ( m ) dt = b ( m − P ( m −
1) + d ( m + 1) P ( m + 1) − [ b ( m ) + d ( m )] P ( m ) . (7)Equation (7) defines a one-dimensional birth-death process with the birth and deathrates given respectively by, b ( m ) = k CR + αk CB exp[ − β ∆ E ( s, m )]1 + exp[ − β ∆ E ( s, m )] , d ( m ) = αk CR + k CB exp[ − β ∆ E ( s, m )]1 + exp[ − β ∆ E ( s, m )] . Its steady-state distribution takes the form, P ( m + 1) = b ( m ) d ( m + 1) P ( m ) = P (0) m (cid:89) i =0 b ( i ) d ( i + 1) . (8)Together with Eq. (6) the full NESS distribution is obtained.Consider the range of ligand concentrations where the receptor is functional, i.e.,exp[ − ∆ E ( m, s )] (cid:28) m = 0 (inactive state favored) and exp[ − ∆ E ( m, s )] (cid:29) m = m (active state favored). Consequently, the ratio b ( m ) /d ( m + 1) changesmonotonically between the limiting values 1 /α and α as m increases from 0 to m .Let m ∗ be the value of m where b ( m ∗ ) /d ( m ∗ + 1) (cid:39)
1. According to Eq. (8), thisis the methylation level where P ( m ) varies slowest with m , i.e., the stationary pointof the distribution. For α < b ( m ) /d ( m + 1) > m < m ∗ ) while b ( m ) /d ( m + 1) < m > m ∗ ). Therefore P ( m ) reaches its peak value at m ∗ . The opposite situation happens for α >
1, where P ( m ) initially decreases with m on the low methylation side, reaches its minimumvalue at m ∗ , and increases on the high methylation side. At α = 1, b ( m ) = d ( m )so that P ( m ) = P (0) b (0) /d ( m ) becomes essentially flat especially when k CR = k CB .The general behavior of the NESS distributions in the two regimes are illustrated in nergy dissipation in an adaptive molecular circuit s at different α values can be found in the Appendix A. m P r o b a b ili t y P (1 , m ) P (0 , m ) P ( m ) (a) m P r o b a b ili t y P (1 , m ) P (0 , m ) P ( m ) (b) Figure 5.
The steady-state distributions P (1 , m ), P (0 , m ) and P ( m ) at m = 4, s = 10 K i . (a) α = 0 .
1; (b) α = α EQ > In the case α <
1, the signal level affects the shape of the distribution by shiftingits peak position m ∗ = m + f ( s ) /e − ( βe ) − ln( k CB /k CR ) (see Appendix A) whichcoincides with the mean methylation level. This is a general feature for adaptationachieved through integral feedback control, i.e., the effect of external signal change isabsorbed by a shift in the average methylation level. According to Eq. (3), the mean receptor activity (cid:104) a (cid:105) in the NESS depends on the signallevel s only through the probabilities for the extreme methylation states at m = 0and m = m . For α < P ( m ) decreases rapidly away from the peak position at m ∗ . Except very close to α = 1 which requires a separate treatment, the adaptationerror as defined by Eq. (4) can be estimated by the largest term in the expression for B . Let m d ≡ min { m ∗ , m − m ∗ } be the distance between m ∗ and the closest extrememethylation state. With the help of Eq. (A.3), we obtain,ln (cid:15) (cid:39) (cid:40) − − α α βe m d , m d < α +1 α − βe ln α ; m d ln α + α − α βe ln α, otherwise . (9)Equation (9) shows that the adaptation error can be decreased by either increasingthe methylation range m or decreasing the parameter α that brings the system furtheraway from equilibrium. At a given α <
1, increasing m allows a greater functionalrange of the receptor and consequently larger values for m d , resulting in an exponentialdecrease of (cid:15) . On the other hand, at a fixed m , decreasing α increases the rate ofexponential decay of (cid:15) . However, when α is below α m = exp( − βe m / (cid:15) m (cid:39) exp( − βe m / nergy dissipation in an adaptive molecular circuit (cid:15) is plotted against α for several different values of m d .
4. Energy dissipation and the ESA trade-off
The nonequilibrium methylation/demethylation dynamics of the MCP receptor requiresenergy input [19]. Within the adaptation model considered here, the rate of energydissipation can be calculated using the standard formula [21, 22, 23]˙ W = 12 β (cid:88) X,X (cid:48) J ( X | X (cid:48) ) ln ω ( X | X (cid:48) ) ω ( X (cid:48) | X ) . (10)Here ω ( X | X (cid:48) ) is the transition rate from state X (cid:48) to state X , J ( X | X (cid:48) ) = ω ( X | X (cid:48) ) P ( X (cid:48) ) − ω ( X (cid:48) | X ) P ( X ) is the net flux from X (cid:48) to X , and P ( X ) is the probabilityfor state X . For our purpose, it is convenient to rewrite the above equation in termscontributions from directed “elementary cycles” [24]. An elementary cycle is a loopformed by nodes and edges of the network that cannot be further decomposed intosmaller loops. Denoting by C l the l th elementary cycle on the network, Eq. (10) can berewritten as ˙ W = 1 β (cid:88) l J ( C l ) A ( C l ) , (11)where J ( C l ) is the probability flux associated with cycle C l , and A ( C l ) = (cid:80) e ∈ C l ln[ ω ( X (cid:48) | X ) /ω ( X | X (cid:48) )], summed along the cycle.For the network model shown in Fig. 2, we define the m th elementary cycle tobe the rectangle between methylation levels m and m + 1, directed counter-clockwiseas indicated by the red arrows. It is simple to verify that the thermodynamic force A ( C l ) = 2 ln[ α EQ /α ] is the same for all cycles. The cycle flux J ( m ) = k CB P (1 , m + 1) − αk CB P (1 , m ) can also be read off easily from the figure. From Eq. (11) we then obtain,˙ W = 2 β (cid:16) ln α EQ α (cid:17) m − (cid:88) m =0 [ k CB P (1 , m + 1) − αk CB P (1 , m )]= 2 k CB β (cid:16) ln α EQ α (cid:17) [(1 − α ) a s − P (1 ,
0) + αP (1 , m )] . With the help of Eq. (3), we obtain finally the following exact expression for the energydissipation in the NESS,˙ W = 2 βτ m (cid:16) − α + B (cid:17) ln α EQ α , (12)where B = αP (0 , − P (1 ,
0) + αP (1 , m ) − P (0 , m ) is a boundary term.Figure 6(b) shows ˙ W against α for selected values of m d . In all cases presented, alogarithmic increase on the far-from-equilibrium side (i.e., α (cid:28)
1) is seen, in agreementwith Eq. (12). Dependence of ˙ W on m d , which enters only through the boundary term B , is essentially negligible. This behavior can be understood from the fact that most nergy dissipation in an adaptive molecular circuit −5 −4 −3 −2 −1 0 110 −10 −8 −6 −4 −2 ln α ǫ / ǫ m d = 1 m d = 3 m d = 5 (a) −5 −4 −3 −2 −1 0 1−10123456 ln α τ m ˙ W m d = 1 m d = 3 m d = 5 (b) Figure 6. (a) Adaptation error and (b) energy dissipation over the methylationtime scale τ m against ln α . Here the ligand concentration is chosen such that m d = m ∗ = m / − m = 4 , of the dissipation takes place in the loop centered around the peak position m ∗ of theNESS distribution P ( m ).In Ref. [19], based on approximate solutions of the adaptation model at m = 4,Lan et al. proposed the Energy-Speed-Accuracy tradeoff relation,˙ W (cid:39) ( c σ a ) ω m ln( (cid:15) /(cid:15) ) (13)to capture the increase in energy dissipation to achieve higher accuracy of adaptationas α is reduced. Here c σ a sets the appropriate energy scale for the problem, and ω m = τ − m . Comparing with our results Eqs. (9) and (12), we see that Eq. (13) needs tobe modified to take into account the dependence of (cid:15) on the distance m d from the actualmean methylation level to the boundaries of the full methylation range, i.e., m = 0 and m = m . In addition, as we see from Fig. 6 , the adaptation error saturates to a valueof the order of (cid:15) m set by m when α falls below α m , while the energy dissipation rate ˙ W keeps increasing. From the calculations presented above, we see that (cid:15) is controlled bythe probabilities for the rare events where the extreme methylation states are visited,while ˙ W is not sensitive to the actual methylation level itself.
5. Phase transition
As we have seen in Sec. 3, there is a qualitative change in the shape of the NESSdistribution P ( m ) at α = 1. For α > P ( m ) is bimodal with peaks at the two endsof the methylation range from 0 to m . The relative weight of the peaks is controlledby the signal strength s . On the other hand, for α < P ( m ) has a single peakin the middle of the methylation range. As the signal strength s varies, the peakposition shifts accordingly but its shape remains more or less the same until either endof the methylation range is reached. As discussed previously by Lan et al. [19], thelatter feature is crucial for the implementation of precise adaptation. In this section we nergy dissipation in an adaptive molecular circuit m d = min { m ∗ , m − m ∗ } → ∞ . For α <
1, Eq. (A.3)shows that the boundary probabilities vanish in this limit, and hence B = 0. On theother hand, for α >
1, this limit implies the activation energies ∆ E (0 , s ) → + ∞ and∆ E ( m , s ) → −∞ . Therefore the receptor is nearly exclusively in the inactive statewhen the methylation level is close to zero, and exclusively in the active state when themethylation level is close to full. Then, the elementary loop current J ( m ) = 0 for all m which in turn yields vanishing dissipation. Here we encounter an interesting examplewhere the detailed balance is violated by the kinetic rates but no dissipation actuallytakes place due to vanishing loop currents. Summarizing, we have in the limit m d → ∞ ,˙ W ∞ = (cid:40) βτ m (1 − α ) ln( α EQ /α ) , < α < , ≤ α ≤ α EQ . (14)The singular behavior of ˙ W against α indicates a true nonequilibrium transition in themodel where the methylation range is infinite.According to Eq. (9), the adaptation error (cid:15) can be made arbitrarily small in theentire adaptive phase α < α c = 1 by increasing m d . On the other hand, ˙ W can bemade arbitrarily small at the same time by choosing an α close to α c . The energydissipation is necessary to generate adaptive behavior, however, there does not appearto be a minimal value for the dissipation rate to support an arbitrarily accurate adaptivesystem.For a system with a finite methylation range, transition between the two phasesis more gradual than what is described above. From Eqs. (A.3) and (A.4), one mayidentify a “correlation length” λ (cid:39) / | ln α | (cid:39) | − α | − . For m d > λ , Eq. (14) canbe directly applied. Corrections need to be considered when m d < λ , based on exactresults derived in previous sections.
6. Conclusions
In this paper, we report a detailed analytical study of the stochastic network modelproposed by Lan et al. shown previously to describe well sensory adaptation in E.coli. To understand this system, we first derive moment equations which are closelyrelated to the rate equations traditionally used to model this type of biological processes.The moment equations implement an integral feedback control scheme at the heart ofthe adaptive behavior. Adaptation in the model is achieved when the nonequilibriumparameter α < α c , where α c = 1 is less than its value α EQ when detailed balance isobserved. By mapping the original ladder network to a one-dimensional birth-deathprocess under the assumption of timescale separation, we obtain analytic expressionsfor the NESS distribution with qualitatively different behavior for α > α < nergy dissipation in an adaptive molecular circuit α ,but approaches a saturated value for α < α m . Interestingly, at a given α , the adaptationerror decreases exponentially with the number of the available methylation states beforethe extreme methylation levels are reached.We also derive an exact formula for the energy dissipation rate by using a cycle-decomposition technique. The energy dissipation rate is found to be insensitiveto the size of the methylation range and also to timescale separation. Althoughour results confirm qualitatively the statement that adaptation within the molecularconstruct that implements integral feedback control requires nonequilibrium driving,there does not appear to be a lower bound on energy dissipation to achieve a givenlevel of adaptation accuracy, in contrary to the Energy-Speed-Accuracy tradeoff relationproposed previously by Lan et al.Although the methylation range of a single MCP receptor is four, we haveinvestigated the behavior of the system with arbitrary methylation range, especiallywhen the methylation range m is large. The extension allows us to examine varioustheoretical issues quantitatively. In the limit m → ∞ , a true nonequilibrium phasetransition at α = 1 can be identified.The current analysis only focuses on the static properties of the system. However,the transient response at short times is also an important component of the molecularadaptive circuit. An understanding of the adaptive behavior in a general setting basedon thermodynamic principles is still lacking. In this respect, lessons may be drawn fromrecent developments in information thermodynamics [18, 25, 26] by viewing the adaptivecircuit as an information processing machine. Acknowledgement
We thank David Lacoste, Kirone Mallick and Henri Orland for helpful discussions. Thework is supported in part by the Research Grants Council of the HKSAR under grantHKBU 12301514.
Appendix A. Approximate expression of the NESS distribution
In this Appendix we derive an approximate analytic expression for the NESS distributiongiven by Eq. (8). Taking the logarithm of the equation, we obtain,ln P ( m + 1) P (0) = m (cid:88) i =0 ln b ( i ) d ( i + 1) = ln d (0) d ( m + 1) + m (cid:88) i =0 φ ( x ( i )) , (A.1)where φ ( x ) = ln 1 + αe x α + e x nergy dissipation in an adaptive molecular circuit x ( m ) ≡ − β ∆ E ( s, m ) + ln( k CB /k CR ) = βe ( m − m ∗ ), with m ∗ = m + f ( s ) /e − ( βe ) − ln( k CB /k CR ). It is straightforward to verify that φ ( x ) is an odd function of x .The function φ ( x ) is well approximated by a piece-wise linear function ψ ( x ) = − ln α, x ≤ − ξ ; α − α +1 x, − ξ < x < ξ ;ln α, x ≥ ξ. (A.2)which has the same slope at x = 0 and same asymptotic values as x → ±∞ . Continuityrequires the choice ξ = α +1 α − ln α . The two functions match each other well except near x = ± ξ .For α <
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