The energy speed accuracy trade-off in sensory adaptation

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1 SUPPLEMENTARY INFORMATION DOI: 1.138/NPHYS2276 The energy speed accuracy trade-off in sensory adaptation 1 The effective potential H(m) In the main text, we introduced a set of F a and F m functions to study the role of energy dissipation in adaptation. F a (a, m, s) = ω a [a G(s, m)], (S1) F m (a, m) = ω m (a a )[β (1 β) m/ω m m G(s, m)], (S2) with a 1 and m m. F a describes the fast (ω a ω m ) relaxation dynamics of the activity a to its steady state value G(s, m). G has opposite dependence on s and m, e.g., s G< and m G> used in this study. Before deriving the effective potential H(m), we briefly explain the form of F m used here. β is a continuous parameter ranging from to 1 which changes the form of F m in order to study both equilibrium and nonequilibrium cases within the same model. To find an equilibrium interaction Fm eq that allows adaptation to a we integrate the detailed balance condition ( 1 a m F a = 1 m a F m ) and use the linear form to impose a to be a steady state (Fm eq (a ) = ). This gives Fm eq = ω m a a (a a ) m G. For non-equilibrium reactions, which break detailed balance but still keep a as a steady state, we use the simple linear form Fm neq = ω m (a a ). Combining the equilibrium and nonequilibrium terms linearly, we have F m = βfm eq + (1 β)fm neq, which is what we have used in this study. Now we derive H(m). First, we can obtain a set of approximate analytical solutions of the probability distribution P (a, m) in the limit where adaptation is much slower than response ω a ω m, where we can separate the probability distribution as P (a, m) = P a (a/m)p m (m). By using this adiabatic approximation, we obtain ( P a (a/m) = exp ω ) a [a G(s, m)] 2, (S3) 2 a where the normalization constant has been absorbed in P m. The distribution P m can also NATURE PHYSICS Macmillan Publishers Limited. All rights reserved.

2 be calculated in this limit by using the flux conservation in the m direction: 1 1 ( J m da = ω m [a a ][β (1 β) m/ω m G ]P a P m m P a m P m ) 1 ( m P a P m m log(p a ) da = e F ω m [a a ][β (1 β) m/ω m G ]P m ω ) a m m P m + m P m [a G]G da a 2π a ( ω m [G a ][β (1 β) ) m/ω m G ]P m m m P m +, ω a from which we have (S4) P m (m) = 1 exp ( H(s, m)), Z (S5) where H(s, m) = ω m (G(s, m) a )[β (1 β) m/ω m G ]dm (S6) m is the effective potential illustrated in Fig. 2a in the main text, where it is abbreviated as H(m). Z is the normalization constant to be determined by 1 m P a (a/m)p m (m)dadm = 1, which gives Z = m 2π e H(s,m) dm. As shown in the main text, the effective potential H(m) can have different minima depending on the value of β. There is always a fixed point at m with fixed activity G(s, m ) = a. However, it is unstable for small β, e.g., for β =. It only becomes stable when 2 H/ m 2 m >, i.e., when β > β c. The critical value β c is given by 2 H/ m 2 m =, which leads to β c = m m G(s, m )/ω m m m G(s, m )/ω m +, as shown in the main text. It is easy to see β c is between and 1. 2 The energy dissipation rate Ẇ For a microscopic system with discrete chemical states, the free energy dissipation rate Ẇ (with thermal energy unit kt = 1) can be written as: Ẇ = AB (J AB J BA ) ln[j AB /J BA ], (S7) where A, B represent the discrete states and J AB, J BA are the transitional rates from A B and B A respectively, see [1] for a recent review and derivations. The interpretation 212 Macmillan Publishers Limited. All rights reserved.

3 of the above equation is clear: J AB J BA is the net current from state A to state B; and ln[j AB /J BA ] is the net energy dissipation for a transition from state A to state B. In equilibrium systems, detailed balance leads to J AB = J BA and therefore Ẇ =. For nonequilibrium systems, detailed balance is broken J AB J BA, and it is easy to show that the energy dissipation rate is positive definite Ẇ >. For the (coarse-grained) continuum model, A and B represent infinitesimal states, each of size da dm, in the phase space. For example, along the a-direction in the phase space, A and B represent two infinitesimal states that are centered at A = (a, m), B = (a + da, m). By discretizing the Fokker-Planck equation, we can obtain the infinitesimal fluxes between them:, J AB = [F a P (a, m)/da + a P (a, m)/da 2 ]dadm J BA = [ a P (a + da, m)/da 2 ]dadm. By plugging the two expressions above in Eq. (S7) and taking the continuum limit da and dm, we obtain the energy dissipation rate for the continuum model used in the main text: Ẇ = 1 m da J 2 a dm[ a P + J 2 m m P ]. This expression of energy dissipation rate (Eq. (S8)) can be derived more rigorously from phase space entropy production rate, as done in [2, 3]. The procedure is to start with the entropy S = k P log(p ) with k the Boltzmann constant, calculate its time derivative Ṡ, identify the positive term which represents the entropy production rate Ṡ prod = k Ji 2 /(P i ), and finally realize that on a steady state this entropy is dissipated to the environment as heat or dissipated energy Ẇ = T effṡprod = kt eff J 2 i /(P i ) where T eff is the effective temperature of the environment. From Eq. (S8), it is clear that the Ẇ is positive definite and vanishes only in equilibrium system where J a = J m =, i.e., the detailed balance condition, is satisfied. 3 The ESA relation To calculate the adaptation error, we can integrate the steady state FP equation in a and m and use the resulting equation 1 da m dm J m =. By using the definition of J m = F m P m m P and Eq. (S2) for F m, the relative adaptive error ϵ = a a /a can be obtained: 1 ϵ = a βω m + 1 β a β 1 1 da m P (a, )da 1 a βω m m 1 m P (a, 4)da, dm m/ω m (a a )G (s, m)p (a, m) ϵ 1 ϵ 2 + ϵ 3, (S8) (S9) 212 Macmillan Publishers Limited. All rights reserved.

4 where compact derivative notation G (s, m) = m G(s, m) will be used from now on. In the above expression for ϵ, the first two integrals (defined as ϵ 1 and ϵ 2 ) constitute the boundary induced errors, and the third one (defined as ϵ 3 ) is the bulk error. We see that the bulk error ϵ 3 vanishes in the fully adaptive model where β = 1. In the following two subsections, we consider the two cases, β = 1 and 1 > β > β c, to derive the Energy- Speed-Accuracy (ESA) relation presented in the main text. 3.1 The ESA relation for β = 1 In this subsection, we study the ESA relation for the biologically relevant case of β = 1, where the bulk error term vanishes (ϵ 3 = ) in Eq. (S9) and we have: ϵ = ϵ 1 ϵ 2 = 1 m [P m (m ) P m ()]e F da m 2π a /ω a P m (m ) P m (). ω m a ω m a By using the solution from the previous section for the case β = 1, we have: ϵ 1 = m ω m a ϵ 2 = m ω m a ( m ( m exp[h(s, m ) H(s, m)]dm) 1, (S1) exp[h(s, ) H(s, m)]dm) 1. (S11) We now apply the steepest descend method around the adaptive minimum at m = m : m e H(s,m) dm e H(s,m ) 2π m ω m G, which leads to: So we have ϵ = ϵ 1 ϵ 2 1 a m G ϵ 1 = 1 a m G 2πω m exp[h(s, m ) H(s, m )], ϵ 2 = 1 a m G 2πω m exp[h(s, m ) H(s, )]. 2πω m e H(s,m ) exp[ H(s, m )] exp[ H(s, )]. (S12) (S13) (S14) For a generic value of s, m is closer to one of the two boundary values m = or m = m. Without loss of generality, we assume m is closer to m, which means that ϵ 1 ϵ 2, though both ϵ 1 and ϵ 2 have the same form (see later). Therefore, we have ϵ 1 a m G 2πω m exp[h(s, m ) H(s, m )] = 1 a m G 2πω m exp[ ω m m m m G(s, m)dm]. (S15) 212 Macmillan Publishers Limited. All rights reserved.

5 Defining ϵ m G /2πω m a 2 and a decay constant d m G(s, m)dm, the final form m of the error is [ ] d ϵ ϵ exp. (S16) m /ω m For m closer to m =, ϵ 2 dominates and the only change for ϵ is that d would be d = m G(s, m)dm. For the fully adaptive model the energy dissipation takes the form Ẇ ω m m /ω m (1 + m/ω m G ) 2. We see from this expression that reducing m /ω m, which reduces ϵ, will at the same time increase the energy dissipation rate. For small adaptation error, the leading order in Ẇ can be approximated as Ẇ ω m m /ω m (S17) By using Eq. S16, we can relate this to the adaptation error, thus arriving at the energyspeed-accuracy (ESA) relation shown in the main text (Eq. 8): Ẇ (c σ 2 a) ω m ln( ϵ ϵ ), (S18) with the explicit expressions for the constants: c = d 1 = ( m G(s, m)dm) 1, σ 2 a =, and ϵ = m G /2πω m a The ESA relation for β c < β < 1 We now study the bulk error ϵ 3 in the adaptive regime 1 > β > β c by expanding the kernel of the integral for ϵ 3 in Eq. (S9) to the second order before applying the steepest descend method: ϵ 3 1 β m /ω m m 2π a a β ω a + [(G(s, m) a )G (s, m)] m=m (m m )P m (m) + 1 ) 2 [(G(s, m) a )G (s, m)] m=m (m m ) 2 P m (m) 1 β β ( m /ω m ) 2 3G 2a (β (1 β) m/ω m G ) 1, ( dm [(G(s, m) a )G (s, m)] m=m P m (m) (S19) Where the derivatives G and G are evaluated at the (adaptive) minima m. It is easy to see from above that the bulk error ϵ 3 becomes smaller as β 1, i.e, as the system becomes 212 Macmillan Publishers Limited. All rights reserved.

6 fully adaptive. By defining a constant e = 3 ( m/ωm)2 a/ω a G /2a, we have β 1/(1 + ϵ 3 /e ) when β is close to 1. To calculate the energy dissipation we first compute the probability currents, which can be obtained using the probability density function P (a, m) obtained above. As the energy dissipation is much smaller in the a direction, we only calculate the current in the m direction: J m (a, m) ω m [a a ][β (1 β) m/ω m G ]P m P m log(p ) = P ( ω m [a a ][β (1 β) m/ω m G ] + m m [F + H]) = P ω m [a a ][β (1 β) m/ω m G ] + P m (G (G a) ω a a + (G a )[β (1 β) m/ω m G ] ω m m ). Note that for the equilibrium case β =, the current flow is zero, in agreement with the plots of the simulation results (Fig. 2c). By using the adiabatic approximation and the steepest descend method, we have Ẇ m dm 1 da J 2 m m P β2 ω m m /ω m (1 + m/ω m G ) 2 (S2) By defining C = a/ω a m /ω m (1 + m/ω m G ) 2, we have the following error energy relation Ẇ C ω m 1 (1 + ϵ 3 /e ) 2. (S21) By writing ϵ 3 = e x and e = e x, and noting that 2 Ẇ / x 2 = at x = x log(2), we find that around ϵ 3 = e /2 the relation above is logarithmic to second order: Ẇ C ω m 1 (1 + e x x ) (x x + log(2)) + O(x x ) 3 (S22) Finally, by defining ϵ,b = e 3/2 e /2, c,b = 8C ω a /27 a, and σa 2 = the variance of the variable a, we have Ẇ (c,b σa) 2 ω m ln( ϵ,b ), (S23) ϵ 3 which has the same form as the ESA relation for the β = 1 case but with different constants c,b and ϵ,b. 212 Macmillan Publishers Limited. All rights reserved.

7 4 The effect of one-step reaction approximation on the energy dissipation rate Consider the full Michaelis-Menten reaction E + S ES E + P with forward rate k f, backward rate k b, production rate k p. For the convenience of computing the energy dissipation rate, we also introduce a counter-production rate k cp from E + P back to ES. The four fluxes can be expressed as: J f = k f [E][S], J b = k b [ES], J p = k p [ES], J cp = k cp [E][P ], where [E], [S], [P ], [ES] are concentrations for enzyme, substrate, product, and enzymesubstrate complex respectively. The net flux J is J = J f J b = J p J cp. The energy dissipation rate W b over the substrate-enzyme binding/unbinding reactions is: W b = (J f J b ) ln(j f /J b ) = J ln(1 + J/J b ). The energy dissipation rate over the production/counterproduction reactions is: W p = (J p J cp ) ln(j p /J cp ) = J ln(1 + J/J cp ). Here, the thermal energy unit kt = 1. The production step is highly irreversible, so J cp J, and J J p = k p [ES]. For Michaelis-Menten reactions, the substrate is in fast chemical equilibrium with the complex, i.e., k b k p, so J/J b J p /J b = k p /k b 1. Finally, the ratio between the two energy dissipation rates is obtained: W b = ln(1 + J/J b) W p ln(1 + J/J cp ) J J b ln(j/j cp ) J J b k p /k b 1. This shows that the energy dissipation rate over the substrate-enzyme binding-unbinding process is negligible in comparison with the energy dissipation rate over the production process in any Michaelis-Menten reactions. Therefore, the one-step reaction approximation used in the paper is valid in determining the energy dissipation rates of the methylation/demethylation processes, which are known to be Michaelis-Menten reactions. 5 The error-energy dissipation relationship in E. coli chemotaxis and its dependence on key system parameters The relationship between the adaptation error (ϵ) and the energy dissipation ( W Ẇ k 1 R ) is shown in Figure 3c in the main text. ϵ = ϵ e α W, W W c, (S24) = ϵ c, W W c. (S25) 212 Macmillan Publishers Limited. All rights reserved.

8 The system operates along the optimum ϵ W line as more energy dissipation reduces adaptation error. The prefactor α in the exponential ϵ W relation (Eq. S24) depends on the ratio of the two kinetic enzymatic rates k R and k B as shown in Fig. S1. In fact, α can be expressed as: α = (1 + k R /k B )/2, as presented in the main text (Fig. 3d) and proven explicitly in the simple case of a 4-state model in this SI. Other than α, the overall ϵ W relationship is characterized by W c and ϵ c. When energy dissipation W is bigger than a critical value W c, the ϵ W curve becomes flat and the adaptation error saturates at a minimum error ϵ c. We define the critical W c to be the energy dissipative rate to achieve an adaptive error within 1% of its saturation value 1 ϵ/ϵ c = 1%. The saturation error ϵ c is caused by the adaptation failure at the methylation boundaries (m =, m = m ). Specifically, For a receptor with m =, CheB-P can not demethylate it any further, and so is the case for CheR with receptors with m = m = 4. Quantitatively, when γ =, i.e., W, the total methylation and demethylation fluxes can be expressed as: m 1 J + = k R m= P (m) = k R [(1 a ) P (m )], m J = K B P 1 (m) = k B [ a P 1 ()], m=1 (S26) (S27) where P a (m) is the probability of the receptor in state (a, m), and we have used the fact the average activity can be expressed as a = m m= P 1(m) and correspondingly 1 a = m m= P (m). In steady state, these two fluxes are equal J + = J, which leads to: a = a + (1 a )P 1 () a P (m ), (S28) from which we obtain the saturation error, i.e., the error at γ = : ϵ c = (1/a 1)P 1 () P (m ). (S29) Here in this section, we describe in detail how W c and ϵ c depends on the key system parameters such as the relative reaction time scales in the activity switching direction (τ a ) comparing to the methylation direction (k 1 R ), and energy gaps at the methylation boundaries ( E() and E(m )). 5.1 The dependence on the activity switching time τ a Here, we study how the speed of activity switching (measured by τ a ) affects adaptation by determining the adaptive error as a function of dissipated energy at τ a = 1, 1, 1,.1,.1k 1 R for ligand concentrations s = 1 2, 1 1 and 1 4 K I (FIG S2a). For each given s, 212 Macmillan Publishers Limited. All rights reserved.

9 the ϵ W performance curves at different τ a values are overlapping until the individual critical saturation point (ϵ c, W c ) is reached. We then varies τ a over a broader range from 1 3 k 1 R to 16 k 1 R. Result clearly shows that slower activity switching speed (larger τ a ) lowers adaptive accuracy (larger ϵ c ) and decreases the critical dissipative energy (smaller W c ). This suggests that slower activity switching limit the ability of the chemotaxis system to use energy for adaptation that makes the ϵ W scaling reaches its saturation faster. When activity switching happens too slow (τ a > 1 3 k 1 R ), the system completely loses its ability to adapt (FIG S2b&c). In addition, slower activity switching increases the percentage of amount of energy dissipated in the activity direction. Under certain condition, this percentage can be more than 5% (as shown in FIG S2d). These results implies that activity switching acts as a valve in the adaptation circuit: when the valve is open (fast switching), two enzymes CheR and CheB can use SAM energy to generate net probability flux to maintain adaptation; however, for large τ a, the system is jammed as the net transitional flux is limited by the slow activity switching. This jamming effect at large τ a increases P (m ) and P 1 (), which compromises the feedback control mechanism and therefore lowers the adaptation accuracy. 5.2 The dependence on the boundary energy differences E() and E(m ) As shown in the main text, the saturation error ϵ c can come from the receptor populations at the two methylation boundary states: (a = 1, m = ) and (a =, m ), where CheB and CheR can not perform their normal feedback actions. Approximately, we have ϵ c = (a 1 1)P 1 () P (m ). Due to fast equilibration in the a direction, we have P (m ) P (m )(1 + exp( E(s, m )) 1, P 1 () P ()(1 + exp( E(s, )) 1. Therefore increasing the energy difference (gap) at the two boundaries (m = and m = m ) can reduce P 1 () and P (m ), and consequently ϵ c. Here, we verify this result by direct simulation of our discrete model. To study the dependence of ϵ c on the intrinsic energy difference of the active and inactive receptor with s =, we denote E(m) as the part of E(s, m) that only depends on m, E(m) is also called the methylation energy. We study the depdence of adaptation accuracy on the boundary methylation energies E() and E(m ). Since the adaptation errors from the two boundaries have opposite signs, for intermediate ligand concentrations, populations at the two boundaries can be comparable, and these two errors could spuriously cancel each other. To get the uncontaminated dependence on an individual boundary energy gap, e.g., m =, we set the other energy gap E(m ) to be essentially infinity to have P (m ) =, and the ligand concentration (s) is chosen so that P () is finite: for the m = boundary, we choose low ligand concentrations (s < K I ), and for the m = m boundary, we choose high ligand concentrations 212 Macmillan Publishers Limited. All rights reserved.

10 (s > K A ). We determine the (ϵ c, W c ) values for each choice of the energy gap. The results are shown in Fig. S3, which shows clearly that increasing E() and E(m ) decreases the saturation error ϵ c exponentially (Fig. S3a&c), and the saturation energy W c increases linearly E() and E(m ) (Fig. S3a&c). 212 Macmillan Publishers Limited. All rights reserved.

11 References [1] Hong Qian. Phosphorylation energy hypothesis: Open chemical systems and their biological functions. Annu. Rev. Phys. Chem., 58: , Oct 27. [2] J. L. Lebowitz and H. Spohn. A Gallavotti-Cohen-type symmetry in the large deviation funcional for stochastic dynmaics. Journal of Statistical Physics, 95(1-2): , [3] T. Tome. Entropy production in nonequilibrium systems described by a Fokker-Planck equation. Brazilian Journal of Physics, 36(4A): , Macmillan Publishers Limited. All rights reserved.

12 Movie Legend Movie S1: Adaptation kinetics in phase space. An animation of the state distribution P (a, m, t) and the corresponding flux field in response to a step increase of s (from s = 1 to s = 1 3 at time=1(a.u.)) is produced with 36 frames in the animated gif format. G = [1 + (s/k d (m))] 1 with K d = K e 2m and K = 1, see Methods in main text for other parameters. 212 Macmillan Publishers Limited. All rights reserved.

13 a 1 Β b 1 Β c 1 Β ε α α α W (kt) W (kt) W (kt) Figure S1: The dependence of α on the k B /k R ratio. k R = 1 is set to be unit to set the time scale. The ϵ W curves are computed for three different choices of k B : a. k B = 2.; b. k B = 1.; c. k B =.5. α can be determined by the slope of the semi-log plot, its values are.75, 1., 1.5 for the three cases shown here, and they agree with the expression α = (1 + k R /k B )/ Macmillan Publishers Limited. All rights reserved.

14 a τ a =1k R τ a =1k R 1 τ a =k R ε 1 1 ε 1 2 ε τ a =.1k R 1 τ a =.1k R 1 3 b s=.1ki W, (kt) 1 4 s=1ki W, (kt) c s=1,k I W, (kt) s=1 2 K I s=1 3 K I ε c W c (k T ) 1 5 s=1 4 K I s=1 5 K I d W a / W total γ = τ a (k 1 R ) τ a (k 1 R ) γ = 1 11 γ = 1 7 γ = τ a (k 1 R ) τ a (k 1 R ) τ a (k 1 R ) τ a (k 1 R ) Figure S2: The effects of receptor activation time τ a. a. ϵ W curves at different signal strength (ligand concentration) s =.1K I, 1K I and 1K I when the receptor activation time varies from τ a =.1K 1 R to τ a = 1K 1 R. At different s, a smaller τ a improves adaptation accuracy as the ϵ W curve saturates at a lower critical error ϵ c along the same performance line. The dependence of the saturation error ϵ c and its corresponding energy dissipation W c on τ a are shown in b&c. ϵ c increases with τ a while W c decreases with τ a. When τ a goes below.1k 1 R, the error does not improve as the system is already in the fast equilibration regime in the a direction. When τ a > 1k 1 R, the system totally loses its ability to adapt due to the slow activation process. d. The percentage of energy dissipated in a direction ( W a ) over the total amount of dissipated energy. This percentage increases when τ a increases. 212 Macmillan Publishers Limited. All rights reserved.

15 a 1 2 b ε c 1 6 W c (kt) 4 35 s=1 3 K I s=1 2 K I s=1 1 K I c E() (kt) d s=1 K I E() (kt) ε c W c (kt) s=1 3 K I s=1 4 K I s=1 5 K I s=1 6 K I E(4) (kt) E(4) (kt) Figure S3: Energy gaps E() and E(4) controls the saturation (minimum) adaptation error ϵ c and the critical dissipative energy W c. a. ϵ c and b. W c versus E() (keeping E(4) = ). c. ϵ c and d. W c versus E(4) (keeping E() = ). Increasing energy gaps E() and E(4) reduces ϵ c exponentially. The system needs to dissipate more energy W c, linearly proportional to E() and E(4), to reach the saturation error ϵ c. Differently colored lines in each panel are for different values of signal strength s. 212 Macmillan Publishers Limited. All rights reserved.