Supplementary Information for

Mechanisms for achieving high speed and efficiency in biomolecular machines

Jason A. Wagoner and Ken A. Dill1

1To whom correspondence should be addressed. E-mail: dilllaufercenter.org

This PDF file includes: Supplementary text Fig. S1 Tables S1 to S4 References for SI reference citations

Jason A. Wagoner and Ken A. Dill1 1 of 13 www.pnas.org/cgi/doi/10.1073/pnas.1812149116 Supporting Information Text 1. Flux for the two-state model

The steady-state flux (number of full cycles per unit time) can be derived from J = PBfm − PArm, where the transition rates are labelled in Figure 1 of the main text, and PA, PB are steady state probabilities that can be solved using the 2 × 2 rate matrix. Here, we give a derivation of the steady state flux that can be used for a more general network of states and for other kinetic properties, like higher moments of flux. The evolution of probability density along the periodic two-state model is

∂P (A , t) j = P (B , t) f + P (B , t) r ∂t j−1 m j c −P (Aj , t) (1 − fc − rm) , [S1] where P (Aj , t) is the probability density of state Aj at time t. The steady state flux can be calculated from the generating functions X −ijk P˜A (k, t) = e P (Aj , t) , [S2] j and similarly for P˜B (k, t). The time evolution of the generating function is

∂P˜A (k, t) X = M P˜ (k, t) , [S3] ∂t An n n∈{A,B} where  −ik  1 − fc − rm rc + fme M = ik [S4] fc + rme 1 − fm − rc, and we are using lettered rather than numbered indices (MAB is M12, etc.). The flux is: ∂ϑ J = i 0 ∂k k=0 f f − r r = c m c m , [S5] fc + fm + rc + rm where ϑ0 is the largest eigenvalue of M (1).

2. Maximizing the machine speed

A. It is better to lower forward barrier heights than to increase reverse barrier heights. Equation 3 of the main text constrains the ratio of forward to reverse rates of a particular transition by ∆µdiss, the total change in basic free energy across the cycle. Any particular change to the basic free energy change across a transition, expressed with the parameter λ in equation 6, may ‡ ‡ affect the forward barrier height gfm, the reverse barrier height grm, or some combination. In this work we focus solely on changes to the forward barrier heights because they have the greatest impact on machine speed. To see this, we can expand the absolute rates to express how changes in free energy will be split (1) between the non-mechanical and mechanical transitions (expressed with the coupling parameter λ as before), and (2) across forward vs. reverse rates (expressed with α):

‡ ‡ gfc = gc − α (1 − λ) ∆µ, ‡ ‡ grc = gc + (1 − α) (1 − λ) ∆µ, ‡ ‡ gfm = gm − αλ∆µ + wδ, ‡ ‡ grm = gm + (1 − α) λ∆µ − w (1 − δ) , [S6]

‡ ‡ where, as in the main text, gc and gm are intrinsic barriers common to both the forward and reverse transitions. We consider α ∈ [0, 1], λ ∈ [0, 1] and, for simplicity, we assume the number of substeps N = 1. The parameters α and λ are agnostic to exact mechanism. These changes in basic free energy may come directly from ∆µ (e.g., ATP hydrolysis), or from up- and down-hill free energy changes of the machine’s conformational cycle, as in the conformationally-driven mechanical step described in the main text. Using these equations for barrier heights, we ask: is it better for changes in basic free energy expressed through α to increase the forward rates (lower the free energy barriers to forward motion) or to decrease the reverse rates (increase the free energy barriers to reverse motion)? First, we write equation S5 as:

J = Π2S/F2S, [S7]

2 of 13 Jason A. Wagoner and Ken A. Dill1 where

Π2S = fcfm − rcrm, [S8]

F2S = fc + fm + rc + rm. [S9]

We find the derivative of J with respect to α at fixed efficiency: ∂J 1 ∂Π Π ∂F = 2S − 2S 2S 2 ∂α ∆µ,w F2S ∂α ∆µ,w F2S ∂α ∆µ,w  1 ∂F  = J ∆µ − 2S F2S ∂α ∆µ,w ≥ 0. [S10]

The last line follows from

∂F2S = (1 − λ) ∆µ (fc + rc) [S11] ∂α ∆µ,w

+λ∆µ (fm + rm) .

Because all rates are strictly positive and λ ∈ [0, 1], 1 ∂F 2S ≤ ∆µ. [S12] F2S ∂α ∆µ,w Inserting equation S12 into the middle line of equation S10 and recalling that ∆µ ≥ 0 shows that ∂J/∂α is strictly positive. Therefore, higher values of α optimize speed, and the unique maximum is found when α = 1: for our two-state model, it is always better for a change in free energy to lower the forward barrier than to increase the reverse barrier. This matches the principles of maximizing machine flux found by Brown and Sivak (2, 3), who also showed that more complex optimization principles may emerge for more complicated models and those with different intrinsic barrier heights. In the main text we focus exclusively on this ideal case when α = 1, where changes in basic free energy affect forward barrier heights with no impact to the reverse barriers, because these are the mechanisms that will have the greatest impact on machine speed. Though we do not explore it here, it is reasonable to expect that some machine mechanisms will perturb the reverse barrier heights.

B. How to optimize forward barriers between the non-mechanical and mechanical steps. We next ask, how should dissipation be spread between the non-mechanical and mechanical transitions? In the main text equations 7, we consider the optimal case (α = 1) where changes to the dissipation across a transition will perturb the forward rate. To optimally parse free energy changes between the non-mechanical and mechanical transitions, we find the value of λ that maximizes speed for the rates given in equations 7, identical to the rates of equations S6 with α = 1. First, we express: ∂J 1 ∂Π Π ∂F = 2S − 2S 2S . [S13] 2 ∂λ ∆µ,w F2S ∂λ ∆µ,w F2S ∂λ ∆µ,w

Noting that ∂Π2S = 0, we multiply throughout by F 2 and solve ∂λ ∆µ,w 2S ∂F 2S = 0. [S14] ∂λ ∆µ,w

The solution to equation S14 gives λopt in equation 9 of the main text.

C. What is the optimal number of mechanical substeps Nopt for a molecular machine?. First, we briefly derive the flux equation for the substep model. We define our model for a machine with N identical substeps to be a machine with 2N total states of alternating chemical and mechanical steps. Because they are identical, every (2-state) substep takes ∆µ/N as input and performs work w/N. Because the model is linear and periodic, we can view this 2N-state cycle as a 2-state cycle with N repeating units. Therefore, the equation for flux through the 2N-state model must match the equation for flux through the 2-state model, divided by N because the full cycle contains N repeating units of the 2-state unit. Therefore J2N (∆µ, w), the 1 flux through a machine with N substeps, is equivalent to N J (∆µ/N, w/N), where J is the flux of the two-state model given in equation S5, and dividing by N accounts for the mean transit time across N repeating units of this two-state unit. ∂J2N We use the substep model just described and solve ∂N = 0 to find Nopt. The value of Nopt depends on the amount of work w and the extent to which the mechanical step is rate-limiting. For example, if the machine has a very large non-mechanical ‡ barrier (a large value of gc ), Nopt will be smaller because there is less value in dividing the (relatively smaller) mechanical barrier. Figure S1 illustrates these two effects and shows that, under most conditions, Nopt is at least 4 and may be much larger. But, there are other practical considerations not included in our theoretical model. First, our model does not include the

Jason A. Wagoner and Ken A. Dill1 3 of 13 Fig. S1. The optimal number of mechanical substeps, Nopt, of a molecular machine. The number of mechanical substeps that maximize speed depends both on the ‡ ‡ amount of work performed and the relative barrier heights of the mechanical (gm) to the non-mechanical (gc ) steps. As shown, a machine with more substeps has the greatest benefit at high work values or when the non-mechanical step is not substantially rate-limiting.

4 of 13 Jason A. Wagoner and Ken A. Dill1 portion of the intrinsic barrier height that would also be split up by the mechanical substeps (e.g., from large conformational changes of the machine). Including this component would increase the benefit of the mechanical substeps. Second, what are the design constraints on reaching a large N? We have shown that N = 4 is evolutionarily accessible by splitting the machine’s cycle between the four components of ATP hydrolysis. It may be difficult for a machine to achieve N > 4, though it is possible that substeps could be obtained from not only the components of ∆µ but also from up- and down-hill transitions in the conformational free energy. Third, these substeps may be applied to not only the mechanical step, but also other important transitions not included in our model. For example, the cytoskeletal walkers , , and have motor heads that undergo cycles of binding and release from their respective cytoskeletal tracks. These individual binding and release steps are themselves gated by the different components of ATP hydrolysis (4–7). Our parameter N only applies to the mechanical step, but this strategy is more general and can split up other important transitions in a machine’s cycle.

3. The in vivo operating conditions of molecular machines Here, we calculate the free energy and work values used in Figure 7 of the main text. We use typical conditions of the ; these machines appear to spend much of their time operating at or near these conditions. These values are also reasonably consistent across the literature, cited below. We calculate the standard and chemiosmotic contributions of input free energy ∆µ and work w using the equations shown in Table S1. The chemiosmotic contributions should be taken with a grain of salt. These quantities should be calculated using activities. Instead, they are calculated using concentrations, which is convenient but less rigorous. In table S2, we list typical cytosolic ion concentrations and membrane potential and we list ∆µ for ATP hydrolysis under various conditions. These quantities and the equations in Table S1 are used below to calculate ∆µ and w for these machines, given in Table S3 and shown in Figure 7 of the main text. cyto Na-K ATPase. For input ∆µ, Na-K ATPase hydrolyzes ATP, ∆µATP. For output work w, Na-K ATPase transports three Na+ ions out of the cell and two K+ ions into the cell, w = 20.3 kT calculated using the cytosolic values listed in Table S2. E. Coli FoF1 ATPase. For input ∆µ, transports 10 protons. Across the membrane, ∆ψ ≈ −125 mV (28–31) and ∆ pH ≈ 0.7 (27–29, 31). For output work w, ATPase synthesizes 3 ATP molecules, where [ATP]/[ADP] ≈ 10, [Pi] ≈ 10 mM (15, 30). For this motor, ∆µ and w values in Table S3 are reported in per-ATP values, not per-cycle values. Animal FoF1 ATPase. For input ∆µ, transports 8 protons (62). Across the mitochondrial membrane, ∆ψ = −150 mV (34, 35, 63) and ∆ pH = 0.5 (34–36). For output work w, ATPase synthesizes 3 ATP molecules, where [ATP]/[ADP] ≈ 2, [Pi] ≈ 50 mM (37, 38). For this motor, ∆µ and w values in Tables S3 are reported in per-ATP values, not per-cycle values. Chloroplast FoF1 ATPase. For input ∆µ, transports 14 protons (64). Across the thylakoid membrane, ∆ψ ≈ −40 mV (15, 55, 58) and ∆ pH ≈ 3.0 (54–57). For output work w, ATPase synthesizes 3 ATP molecules, where [ATP]/[ADP] 0 ≈ 2.4 (23, 24), [Pi] ≈ 2 mM (25). This value of w = 7.1 kT also matches the values from Refs. (20–22), though the exact measured concentrations vary. For this motor, ∆µ and w values in Table S3 are reported in per-ATP values, not per-cycle values. cyto The Plasma-membrane channel (PMCA). For input ∆µ, PMCA hydrolyzes ATP, ∆µATP. For output work w, PMCA transports one Ca2+ out of the cell, w is calculated using the cytosolic conditions of Table S2. cyto The Sarco/Endoplasmic reticulum calcium ATPase (SERCA). For input ∆µ, SERCA hydrolyzes ATP, ∆µATP. For output work w, SERCA counter-transports two Ca2+ into the Sarco/Endoplasmic reticulum and two protons out (65). 2 2 [Ca +]SR / [Ca +] ≈ 10, 000 (39, 40, 42). It is generally assumed that there is no pH gradient (41, 43) and that ∆ψ ≈ 0 (42), though this may be incorrect (43, 65). chloro The plasma membrane proton pump (PM H+ pump). For input ∆µ, hydrolyzes ATP, ∆µATP . For output work w, transports one proton out of the chloroplast stroma. Across the membrane, ∆ψ ≈ −120 to −160 mV (46, 51) and ∆ pH ≈ 1.5-2.0 (46, 52, 53). muscle Myosin II. For input ∆µ, hydrolyzes ATP, ∆µATP . For output work w, generates ≈ 6 pN of force against ≈ 6 nm step (4, 60, 61). These numbers are consistent over a wide range of varying tension applied to muscle (4, 60, 61). 0 The proton pump (H+ PPi-pump). For input ∆µ, hydrolyzes pyrophosphatase, ∆µ ≈ 11.13 kT (16), [PPi] ≈ .2-.3 mM (44) and [Pi] ≈ 1-2 mM (25). For output work w, transports one proton into a vacuole from the chloroplast stroma. Across the membrane, ∆ψ ≈ −20 mV (20, 46, 47) and ∆ pH ≈ 2.5 (45, 47–49). chloro The chloroplast V-ATPase. For input ∆µ, hydrolyzes ATP, ∆µATP . For output work w, transports two protons into a vacuole from the chloroplast stroma; w components are calculated as for the H+ PPi-pump. The sodium-calcium exchanger (NCX). NCX transports 3 Na+ downhill in exchange for 1 Ca2+ uphill. We study NCX under its role in the cardiac action potential, one of its central functions where it rapidly clears Ca2+ from the cell. The conditions (concentrations and ∆ψ) vary during the action potential, but the bulk of NCX function occurs while ∆ψ ≈ −90 mV, 2+ 2+ close to resting conditions (59). It is more difficult to estimate concentrations; we use the rough estimate [Ca ]out/[Ca ]in ≈ 10 during this phase of the action potential and use the cytosolic concentrations given above for Na+. For input ∆µ, NCX transports 3 Na+ downhill. ∆ψ ≈ −90 mV (59), and we use the cytosolic concentrations from Table S2. For output work w, 2+ 2+ 2+ NCX transports 1 Ca uphill. ∆ψ ≈ −90 mV (59), [Ca ]out/[Ca ]in ≈ 10. The sodium-calcium-potassium exchanger (NCKX). NCKX pumps 4 Na+ downhill for 1 Ca2+ and 1 K+ uphill. 2+ We study NCKX under one of its principal roles in the retinal rod cell, where ∆ψ ≈ −40 mV and [Ca ]in ≈ 500 nm (50). As

Jason A. Wagoner and Ken A. Dill1 5 of 13 Type of input free energy Example machine ∆µ0 ∆µ0 ATP hydrolysis Myosin II ∆µ0 kTln [ATP] ATP [ADP][Pi ] m+ [Xout ] Inward flux of ion X FoF1-ATPase, NCX −mF∆ψ kTln [Xin ] Type of Output work Example machine w0 w0 0 [ATP] ATP synthesis FoF1-ATPase ∆µ kTln ATP [ADP][Pi ] Outward flux of ion Xm+ NCX −mF∆ψ kTln [Xout ] [Xin ] Step d against constant force f Myosin II fd - Table S1. Different sources of input free energy and output work for molecular machines. The total free energy has standard ∆µ0 and chemiosmotic ∆µ0 components, and similarly for work. A membrane potential ∆ψ is defined relative to outside.

6 of 13 Jason A. Wagoner and Ken A. Dill1 Condition value ∆ψ -70 mV + [Na ]in 5 mM (8–10) + [Na ]out 150 mM (9, 11–13) + [K ]in 5 mM (9, 11–13) + [K ]out 200 mM (8, 10, 13) 2 [Ca +]in 100 nM (9, 14) 2 [Ca +]out 2 mM (11–14) 0 Standard concentrations, ∆µATP 12.8 kT (15–17) cyto Cytosol, ∆µATP 23.0 kT (18, 19) chloro Cholorplast stroma, ∆µATP 19.9 kT (15, 20–25) muscle Resting , ∆µATP 25.8 kT (26) Table S2. Resting cytosolic membrane potential and ion concentrations and the free energy of ATP hydrolysis ∆µ under different conditions. 0 [ATP] We calculate ∆µ = µATP − µADP − µP , or ∆µ = ∆µ + kTln . i [ADP][Pi]

Jason A. Wagoner and Ken A. Dill1 7 of 13 Machine ∆µ (kT) w (kT) η

E. Coli FoF1- ATPase 21.6 (15, 27–33) 19.7 (15–17, 30) 0.91

Animal FoF1- ATPase 18.64 (34–36) 16.5 (15–17, 37, 38) 0.89 Na-K ATPase 23.0 (18, 19) 20.3 0.88 SERCA 23.0 (15–19) 18.4 (39–43) 0.80

Proton PPi Pump 8.8 (16, 25, 44) 6.5 (20, 45–49) 0.74 NCKX 23.7 (50) 16.7 (50) 0.70 PMCA 23.0 (15–19) 15.4 (14) 0.67 V ATPase 19.9 (15–17, 20–25) 13.1 (20, 45–49) 0.66 PM H+ pump 19.9 (15–17, 20–25) 11.7 (46, 51–53) 0.59

Chloroplast FoF1- ATPase 39.5 (15, 54–58) 19.9 (15–17, 20–25) 0.50 NCX 20.7 (59) 9.3 (59) 0.45 Myosin II 25.8 (15–17, 26) 8.8 (4, 60, 61) 0.34 Table S3. The in vivo input-output operating conditions of molecular machines. Input chemical potential ∆µ, output work w, and the efficiency η = w/∆µ of molecular machines in vivo.

8 of 13 Jason A. Wagoner and Ken A. Dill1 with NCX, we approximate other concentrations by their cytosolic values. For input ∆µ, NCKX transports 4 Na+ downhill. ∆ψ ≈ −40 mV (50), and we use the cytosolic concentrations from Table S2. For output work w, NCKX transports 1 Ca2+ + 2+ and 1K uphill. ∆ψ ≈ −40 mV (50), [Ca ]in ≈ 500 nm (50), and for all other concentrations we use the cytosolic values in Table S2.

4. In vitro fits

A. Fitting procedures. We fit models to in vitro force-velocity data for kinesin (66), F1-ATPase (67), dynein (68), RNA polymerase (RNAP) (69), Na-K ATPase (70), and myosin V (71). We fit a single force-velocity curve for each of these machines to the two-state model of Figure 1 using rates from equations 7 of the main text. For F1-ATPase, we use this two-state model with three identical substeps (6 total transitions). Setting N = 3 is necessary to get a compatible fit to the data, and also matches the three catalytic dwells observed from single molecule experiments (72–74). Because the substeps are identical, there are no extra parameters. For kinesin, dynein, RNAP, F1-ATPase, and myosin V the data are reported as velocity V with respect to applied force f. We calculate the work w = fd and velocity V = Jd., where d is the step size of the motor. For kinesin, dynein, and RNAP, ∆µ is approximated from the stall force rather than the experimental conditions of ∆µ. This is because these motors stall below 100% efficiency. Kinesin, for example, has an ATP-driven backstep that lowers its stall force. Our model can capture these effects with a simple extension that includes an ATP-driven backstep, but we do not include these analyses here since it would introduce more parameters. Our two-state model provides compatible fits to the data without including these additional pathways. For Na-K ATPase, the pump current I is reported with respect to applied voltage ∆ψ (relative to outside of the mem- brane) (70). This machine pumps three Na+ ions out of the membrane and two Cl− ions in. We calculate the work:

 3 2  [Na]out [Cl]in w = −F∆Ψ + kT ln 3 2 , [S15] [Na]in [Cl]out where F is the Faraday constant. For each machine, best fit parameters are determined by minimizing the chi-squared error weighted by the inverse experimental error using the LMFIT package (78). We find that single fits do not typically converge to the global optimum. We perform a global optimization by randomly selecting 200 initial parameter values. All results converged to the global optimum within 50 iterations.

B. Results. In Figure 8 of the main text, the model curve and the force-velocity data for each machine is translated to speed- efficiency by calculating η = w/ |∆µ|, where w is calculated as described above and ∆µ is calculated from the experimental conditions. Fit parameters are given in Table S4. Myosin V and Na-K ATPase are not included because we do not aim to infer parameters for these machines; we aim only to plot the speed-efficiency behavior. We do not aim to infer parameters for myosin V because there are very few data points with large experimental errors (71). We do not aim to infer parameters for Na-K ATPase because the complexity of the machine is beyond the scope of this manuscript, detailed models based on the Post-Albers mechanism (79, 80) have been studied by others (75–77). The fit parameters must be taken with a grain of salt. These models are more simple than the true reaction mechanisms of these machines (kinesin, for example, has an ATP-driven backstep not included here). The data are well-fit by our simple model, and increasing model complexity leads to more parameters and to over-fitting. This dilemma is best handled by accumulating data from multiple sources to constrain a model of sufficient complexity, as in the recent work of Sumi for kinesin (81). This procedure is beyond the scope of this work and must be taken on a machine-to-machine basis.

Jason A. Wagoner and Ken A. Dill1 9 of 13 −1 −1 Machine N kc (s ) km (s ) λ δ Na-K ATPase (70) many (75–77) - - - - Myosin V (71) 3 (6) - - - - Kinesin (66) 1 67.15 3.66 ×10−2 0.81 0.15

F1-ATPase (67) 3 (73, 74) 42.84 0.859 1.0 0.7 Dynein (68) 1 832.0 2.16 ×10−2 1.0 0.45 RNAP (69) 1 2.07 ×10−5 0.679 0.22 1.0 Table S4. Best-fit parameters and number of mechanical substeps N for the six machines studied here. See text for a description of data and fitting procedures. Values of N are taken from experimental observation.

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