The nonlinear chemo-mechanic coupled dynamics of the F₁-ATPase molecular motor

Abstract

The ATP synthase consists of two opposing rotary motors, F₀ and F₁, coupled to each other. When the F₁ motor is not coupled to the F₀ motor, it can work in the direction hydrolyzing ATP, as a nanomotor called F₁-ATPase. It has been reported that the stiffness of the protein varies nonlinearly with increasing load. The nonlinearity has an important effect on the rotating rate of the F₁-ATPase. Here, considering the nonlinearity of the γ shaft stiffness for the F₁-ATPase, a nonlinear chemo-mechanical coupled dynamic model of F₁ motor is proposed. Nonlinear vibration frequencies of the γ shaft and their changes along with the system parameters are investigated. The nonlinear stochastic response of the elastic γ shaft to thermal excitation is analyzed. The results show that the stiffness nonlinearity of the γ shaft causes an increase of the vibration frequency for the F₁ motor, which increases the motor’s rotation rate. When the concentration of ATP is relatively high and the load torque is small, the effects of the stiffness nonlinearity on the rotating rates of the F₁ motor are obvious and should be considered. These results are useful for improving calculation of the rotating rate for the F₁ motor and provide insight about the stochastic wave mechanics of F₁-ATPase.

Introduction

The ATP synthase consists of two opposing rotary motors, F₀ and F₁, coupled to each other. The F₁ motor consists of a hexamer of α and β subunits with stoichiometry α₃β₃ surrounding a central cavity containing the γ subunit which is a bent coiled-coil. The largest portion of F₀ comprises 12 c subunits. These 12 subunits are assembled into a trans-membrane disk. The F₁ motor is fueled by nucleotide hydrolysis and drives the γ shaft clockwise (viewed from F₁ toward the membrane), whereas the F₀ motor is fueled by the ion-motive force and drives the γ shaft counterclockwise [1–4]. When the F₁ motor is not coupled to the F₀ motor, it can work in the direction hydrolyzing ATP, as a nanomotor called F₁-ATPase. Rotation of the γ subunit in an isolated F₁ during ATP hydrolysis has been demonstrated experimentally [5–9].

To understand the mechanism of the F₁-ATPase motor, several insightful models have been proposed [10–15]. Yasuda et al. found that the 360° rotation cycle of the γ subunit includes three repeated 120° rotation steps which are further resolved into two substeps of 30° and 90°, and the probability density peaks of finding the γ subunit at some angle correspond to 90°, 120°, 210°, 240°, 330° and 360°. By electrostatic interaction between the β subunits and the coiled coil of the γ subunit, six potential wells are formed between subunits γ and α₃β₃ [8, 9]. Oster et al. proposed that ATP binding causes elastic strain at the catalytic site and binding of a substrate to the β subunit causes its conformational change (elastic strain) which drives the γ shaft rotation anticlockwise by the electrostatic interactions between γ protrusion and β subunit [16]. The elastic strain potential was considered to be a linear function of the rotational angle of the γ shaft [17]. The characteristic length scales of the F₁-ATPase are about 10 nm and their motion is a stochastic phenomenon which is well observed in the rotation angle of the γ shaft which undergoes small fluctuations due to the random collisions with solvent molecules [8]. The random collisions can induce a random torque. Based on these studies, the authors proposed a stochastic wave mechanics of F₁-ATPase by which a linear chemo-mechanical coupled model for F₁-ATPase was given [18].

However, the mechanics of the protein and DNA has been investigated [19–22]. The force-deformation relationship of the protein is similar to that of DNA. The stiffness of the protein changes along with increasing the force applied to it [20–22]. The nonlinearity has an important effect on the natural frequency of the F₁-ATPase. It will influence the rotating rate of the F₁-ATPase. Therefore, a nonlinear chemo-mechanical coupled dynamic model for F₁-ATPase should be developed.

In this paper, considering the nonlinearity of the γ shaft stiffness for the F₁-ATPase, a nonlinear chemo-mechanical coupled dynamic model of F₁ motor is proposed. Nonlinear vibration frequencies of the γ shaft and their changes along with the system parameters are investigated. The nonlinear stochastic response of the elastic γ shaft to thermal excitation is analyzed. The results show that the stiffness nonlinearity of the γ shaft causes an increase of the vibration frequency for the F₁ motor which increases the motor’s rotation rate. When the concentration of ATP is relatively high and the load torque is small, the effects of the stiffness nonlinearity on the rotating rates of the F₁ motor are relatively obvious and should be considered.

The chemo-mechanic coupled dynamic equations

Figure 1a shows a view of an electron density map of the F₁–c₁₀ complex at 3.9 Å resolution [23]. Figure 1b is the sketch of the F₁ motor. From the figures, it is known that the γ subunit can be considered as an elastic shaft and the γ protrusion at one end of the γ subunit is considered as a lump inertia J. The other end of the γ subunit is fixed within the potential well. Setting the origin of the x axis at the center of the potential well, l is the distance from the origin to the center of the structure with lump inertia J. Thus, the dynamic wave model of the γ subunit can be given as shown in Fig. 1c (here, k is stiffness of the potential well).

Fig. 1.

Electron density map of F₁ and its dynamic model. a A view of an electron density map of the F₁–c₁₀ complex at 3.9 Å resolution. b Sketch of the F₁ motor c Dynamic wave model of the γ subunit. Figure 1a is reprinted from [23], with permission from AAAS

The stochastic twisting wave of the elastic γ shaft can be described by its dynamic equation

1

with the dynamic viscous torque (here, θ(x,t) is the dynamic twisting angle displacement of the γ shaft, x is the coordinate of one point on the γ shaft, t is time, ζ is the drag coefficient between the γ shaft and water); dynamic elastic torque (here, G is the shear modulus of elasticity, I_p is the rotary inertia of the γ shaft); inertia torque (here, ρ is density of the protein); τ_flunct is the Brownian torque from thermal fluctuations. It is taken as a Gaussian white noise related to drag coefficient ζ₂ according to the fluctuation–dissipation theorem

2

3

where ζ₂ is the friction coefficient between the γ protrusion and water, k_B is the Boltzmann constant, and T is the temperature.

Substituting the related torques into 1, the stochastic dynamic equation of the elastic γ shaft can be given as

4

The shear modulus of elasticity G of the γ shaft is nonlinear and can be expressed as

5

where G₀ is the linear shear modulus of elasticity of the γ shaft, b is a coefficient, θ_T is the twisting angle displacement of the γ shaft, θ_T = θ₀ + θ, θ₀ is the static twisting angle displacement of the γ shaft.

The static twisting angle displacement θ₀ can be calculated by

6

where τ_s is the static torque applied to the γ subunit, τ_s = τ_drive − τ_load , τ_drive is the driving torque applied to the γ the subunit, which equals the derivative of the membrane potential with respect to the γ subunit position angle, (here, Δμ_e is the work done in hydrolyzing one ATP). It is a chemo-mechanic coupled term which represents the influence of the chemical energy on mechanical motion. τ_load is load torque applied to the γ the subunit (trans-membrane ion gradient, torque from laser trap, or torque from electromagnet).

Set small parameter , and then Eq. 5 can be changed into

7

Substituting Eq. 7 into Eq. 4 yields

8

Neglecting high-order terms, we find

9

where and .

Substituting θ_T = θ₀ + θ into Eq. 9, yields

10

Equation 10 is just the chemo-mechanic coupled dynamic equation. θ₀ is the chemo-mechanic coupled term dependent on the chemical potential across the membrane, which represents the effects of the chemical energy on mechanical motion.

The solution of the chemo-mechanic coupled dynamic equation

Neglecting the damping term and the exciting torque, substituting into Eq. 10, yields

11

where and , ϕ(x) is the mode function of the γ shaft.

Let Eq. 11 equal constant , thus

12

13

where .

The solution of Eq. 12 can be written as

14

Thus, the boundary conditions of the γ subunit can be given by and . Substituting Eq. 14 into the boundary conditions yields

15

From Eq. 15, the natural frequency of the linear vibration for the γ subunit can be obtained. Substituting the natural frequency into 14, the mode function can be given as

16

where ω_i is the ith-order natural frequency of the γ subunit, and ϕ_i(x) is the ith-order mode function.

Using the Lindstedt-Poincaré method, one assumes

17

and

18

where ω is the nonlinear vibration frequency.

Substituting Eqs. 17 and 18 into 13, and equating coefficients of like powers of ε on both sides, one obtains

19a

19b

19c

Here, the initial conditions are

20

The solution of Eq. 19a under the above initial conditions is

21

Substituting Eq. 21 into Eq. 19b yields

22

In order to remove the secular term, i.e., that which grows without bound, let the coefficient of cos ωt be zero, and then

23

Thus

24

Substituting Eqs. 21 and 24 into (19c) yields

25

where and .

In order to remove the secular term, let coefficient of the cos ωt to be zero, and then

26

Thus

27

Substituting Eqs. 21, 24, and 27 into (17) yields

28

Substituting Eq. 14 and 28 into , the nonlinear free vibration displacements of the γ shaft can be obtained. Substituting Eqs. 23 and 26 into (18), the nonlinear vibration frequency of the γ shaft can also be given as

29

Stochastic response of the elastic γ shaft to thermal excitation

Linear stochastic response

After mode functions are obtained, the stochastic response of the elastic γ shaft to thermal excitation can be resolved. The solution of Eq. 10 can be given in the following form:

30

Substituting Eq. 30 into (10), letting ε = 0, and using the orthogonality of the mode function, yields

31

where q_i(t) is the generalized coordinate of the γ subunit wave, Q_i(t) is the generalized force, and ξ_i is the relative damping coefficient.

The generalized force is

32

As the thermal excitation applied to the γ protrusion has the most important influence on the stochastic wave of the γ subunit, the fluctuating torque can be written as

33

Substituting Eq. 33 into (32) yields

34

Thus, the RMS value of the γ subunit wave at any point x_p can be given as

35

where H_i(ω) is the complex frequency response function between the thermal excitation and the twisting wave of the γ subunit.

From Eq. 10, the complex frequency response function can be deduced:

36

Substituting Eq. 36 into (35) yields

37

The rotating rate of the γ subunit is mainly decided by the wave of the partial subunit within the potential well. As one end of the potential well, x = 0. Hence, the RMS value of the γ subunit wave at point x = 0 is

38

In Eq. 38, the wave amplitude of the lowest mode of the γ subunit is the maximum. The lowest mode wave plays the main role for escape of the γ subunit from the potential well.

Let σ_θ denote the root mean square error of the wave for the γ subunit at its one end (within potential well, x = 0). Then, the root mean square error can be calculated from Eq. 38 easily.

Nonlinear stochastic response

Based on the linear stochastic response of the elastic γ shaft to thermal excitation, the nonlinear stochastic response can be obtained as well. Substituting Eq. 30 into (10), using the orthogonality of the mode function yields

39

Let g(q_i(t)) denote the nonlinear term of Eq. 39, thus

40

The equivalent linear equation for Eq. 39 can be given in the following form

41

where ξ_e and k_e are the equivalent damping and stiffness coefficients, respectively.

The difference between Eq. 39 and 41 is

42

In order to determine coefficients ξ_e and k_e, the following conditions should be met:

43a

and

43b

where E(e²) is the variance of the function .

Substituting Eq. 42 into (43a) and (43b) yields

44

From Eq. 44, the coefficients ξ_e and k_e can be obtained:

45

The thermal fluctuations are considered as steady Gaussian white noise, thus

46

Thus, Eq. 45 can be changed into the following form:

47

For the steady Gaussian white noise, can be given by

48

where , and gives the variance of the linear system.

Substituting Eq. 40 into (47) yields

49

Combining Eq. 48 with 49 yields

50

Each term of Eq. 50 is multiplied by , giving

51

where and are the wave variance of the dynamic angle at point x_p for linear and nonlinear systems, respectively.

From Eq. 51, one obtains

52

where .

Equation 52 can be written as

53

The wave variance of the dynamic angle at point x_p = 0 can be given as

54

The rotation rate of the γ subunit

The rotation rate of the γ subunit can be given as

55

where t_1/3 denote a 120° rotation time of the γ subunit, . t_mech is the time of the γ subunit for 120° mechanical step, , consisting of 90° and 30° mechanical substep times; here, and . t_reac is the reaction time of the ATPase, . t_coup is the coupled time of the γ subunit rotation and ADP, P_i dissociation, . [T], [D], and [P] are the concentration of ATP, ADP, and P_i, respectively; k_h is the rate of 90° rotation of the γ subunit of the F₁-ATPase, k′_h is its rate of 30° rotation; k_x is the binding rate constant for species x, k_− x is the corresponding dissociation rate constant (x = T, D, or P).

The static balance equation of the γ shaft rotation can be described by

56a

56b

where k is the stiffness of the potential well, and θ_end is the shift amount of the partial γ subunit within the potential well. φ is the rotation angle of the γ subunit, and ζ is the friction coefficient between the actin filament or the bead and the water.

From Eq. 56a, the shift amount θ_end of the partial γ subunit within the potential well can be given. Thus, the probability density function p (θ) of the γ subunit wave can be determined and the probability P_α of the γ subunit escape from the potential well can be calculated (, where α is the width of the potential well). So, the oscillation time t_α of the γ subunit within the potential well before escape is

57

where ω₁ is the lowest vibration frequency of the γ subunit.

Let β denote the angular distance between the two potential wells. Then, from Eq. 56b, the jumping time from one well to another can be obtained:

58

As state above, a 120° rotation has been resolved into two substeps of approximately 30° and 90°, which correspond to two potential wells. Thus, the times for the 30° and 90° substeps are

59a

and

59b

The nonlinear vibration of the γ subunit decides the lowest vibration frequency ω₁ and the root-mean-square error of the γ subunit wave. Thus, it has an important effect on the rotation rate of the F₁ motor.

Results and discussion

The above equations are utilized for the analysis of the free vibration of the γ subunit. The parameter values of the motor system are given in Table 1. Figures 2, 3, 4, and 5 show changes of the first three vibration frequencies along with system parameters. Tables 2, 3, 4, and 5 give changes of the difference between the nonlinear and linear vibration frequencies along with system parameters. From these results we can conclude:

1. The vibration frequencies of the γ subunit grow with increasing nonlinear parameter ε. This is because the nonlinear stiffness of the γ subunit is larger than the linear stiffness.

As the chemical torque τ_drive (written as τ at below) on the γ subunit grows, the natural frequencies of the linear vibration do not change, but the nonlinear free vibration frequencies grow. For larger nonlinear parameter ε, the nonlinear free vibration frequencies grow more obviously with with an increase in the chemical torque.

Table 3.

The difference between the nonlinear and linear frequencies for various diameters d

d (nm)		1.5	2	2.5	3	3.5
Δω1	ε = 0.1	0.97 × 109	1.05 × 109	0.92 × 109	0.88 × 109	0.84 × 109
	ε = 0.25	2.38 × 109	2.32 × 109	2.19 × 109	2.11 × 109	1.94 × 109
	ε = 0.5	4.46 × 109	4.40 × 109	4.08 × 109	3.82 × 109	3.49 × 109
Δω2	ε = 0.1	1.90 × 109	2.00 × 109	2.30 × 109	1.90 × 109	1.30 × 109
	ε = 0.25	5.30 × 109	5.20 × 109	4.90 × 109	3.50 × 109	3.90 × 109
	ε = 0.5	8.80 × 109	8.90 × 109	8.70 × 109	7.60 × 109	7.00 × 109

Table 4.

The difference between the nonlinear and linear frequencies for various values of the stiffness k

k(pNnm/rad)		175	200	225	250	275
Δω1	ε = 0.1	0.91 × 109	1.41 × 109	0.92 × 109	0.90 × 109	0.95 × 109
	ε = 0.25	2.21 × 109	2.23 × 109	2.19 × 109	2.15 × 109	2.32 × 109
	ε = 0.5	4.00 × 109	4.10 × 109	4.08 × 109	4.21 × 109	4.24 × 109
Δω2	ε = 0.1	1.60 × 109	2.40 × 109	2.30 × 109	2.50 × 109	1.40 × 109
	ε = 0.25	3.90 × 109	4.50 × 109	4.90 × 109	4.40 × 109	4.40 × 109
	ε = 0.5	7.90 × 109	8.10 × 109	8.70 × 109	8.70 × 1010	8.10 × 109

Table 5.

The difference between the nonlinear and linear frequencies for various values of l

l(nm)		3.5	5.5	7.5	9.5	11.5
Δω1	ε = 0.1	1.8 × 109	1.1 × 109	9.2 ×108	6.7 × 108	6.1 × 108
	ε = 0.25	4.30 × 109	2.80 × 109	2.19 × 109	1.74 × 109	1.46 × 109
	ε = 0.5	8.00 × 109	5.30 × 109	4.08 × 109	3.29 × 109	2.78 × 109
Δω2	ε = 0.1	3.7 × 109	2.5 × 109	2.3 × 109	1.4 × 109	0.9 × 109
	ε = 0.25	9.00 × 109	5.90 × 109	4.90 × 109	3.50 × 109	2.60 × 109
	ε = 0.5	1.66 × 1010	1.10 × 1010	8.70 × 109	6.60 × 109	6.20 × 109

Table 1.

The parameter values of the motor system

l(nm) 7.5	d(nm) 1.25	D(nm) 2.75	h(nm) 5	G(N/m2) 5 × 107	k(pN nm/rad) 225
ρ (kg/m3)	kT (M − 1 s − 1)	k(M − 1 s − 1)	k(M − 1 s − 1)	k(M − 1 s − 1)	k(M − 1 s − 1)
1000	1.8 × 107	5 × 105	2 × 104	6 × 102	2 × 103

Fig. 2.

Changes of the nonlinear vibration frequency along with chemical torque τ_drive. a mode 1 b mode 2

Fig. 3.

Changes of the nonlinear vibration frequency as a function of the γ shaft diameter d. a mode 1 b mode 2

Fig. 4.

Changes of the nonlinear vibration frequency along with stiffness k of the potential well. a mode 1 b mode 2

Fig. 5.

Changes of the nonlinear vibration frequency along with length l of the γ shaft. a mode 1 b mode 2

Table 2.

The difference between the nonlinear and linear frequencies for various torques τ(pNnm)

τ(pNnm)		25	29	33	37	41
Δω1	ε = 0.1	0.74 × 109	0.82 × 109	0.92 × 109	1.04 × 109	1.13 × 109
	ε = 0.25	1.71 × 109	2.00 × 109	2.19 × 109	2.49 × 109	2.68 × 109
	ε = 0.5	3.22 × 109	3.71 × 109	4.08 × 109	4.60 × 109	4.96 × 109
Δω2	ε = 0.1	1.80 × 109	2.00 × 109	2.30 × 109	2.50 × 109	2.80 × 109
	ε = 0.25	3.30 × 109	4.20 × 109	4.90 × 109	5.40 × 109	5.80 × 109
	ε = 0.5	6.60 × 109	7.80 × 109	8.70 × 109	9.70 × 109	1.05 × 1010

As the chemical torque τ on the γ subunit grows, the relative errors between the nonlinear and linear free vibration frequencies increase. This is because the nonlinearity of the γ subunit increases with the chemical torque τ. As the order number of the mode increases, the effects of the chemical torque τ on the nonlinear free vibration frequencies become small.

For mode 1 and ε = 0.25, the relative errors between the nonlinear and linear free vibration frequencies are 0.9% at τ = 25 pNnm, 1.1% at τ = 33 pNnm, and 1.3% at τ = 41 pNnm. For mode 1 and ε = 0.5, the relative errors between them are 3.7% at τ = 25 pNnm, 4.6% at τ = 33 pNnm, and 5.5% at τ = 41 pNnm.

Hence, the effects of the chemical torque on the nonlinear free vibration frequencies are relatively obvious and cannot be neglected for relatively large nonlinear parameter ε.

1. As the diameter d of the γ subunit grows, the linear and nonlinear free vibration frequencies drop. As the diameter d grows, the relative errors between the nonlinear and linear free vibration frequencies drop for mode 1 and ε = 0.25 or 0.5 (here, the system nonlinearity is relatively strong). This shows that the effect of the diameter d on the nonlinearity of the system drops gradually. For mode 1 and ε = 0.1 or higher order modes (here, the system nonlinearity is relatively weak), the relative errors between the nonlinear and linear free vibration frequencies first grows, reaches a maximum value, and then drops with increasing diameter d.

2. As the stiffness k of the potential well grows, the linear and nonlinear free vibration frequencies grow. As the stiffness k grows, the relative errors between the nonlinear and linear free vibration frequencies change. This shows that the stiffness k has an effect on the nonlinearity of the system as well. As the stiffness k change, the relative errors between the nonlinear and linear free vibration frequencies reach relatively large values at some stiffness k which corresponds to relatively large system nonlinearity.

3. As the length l of the γ subunit grows, the linear and nonlinear free vibration frequencies drop, and the relative errors between the nonlinear and linear free vibration frequencies decrease. This is because the nonlinearity of the γ subunit decreases with the length l.

Changes of the mechanical step time for the γ subunit rotation along with the nonlinear parameter ε are given in Table 6 (here, τ_drive = 45 pN nm and τ_load = 0). Table 7 gives changes of the relative error of the mechanical step time between the nonlinear and linear systems. These results show:

1. When the stiffness nonlinearity of the γ subunit is considered, the mechanical step time for the γ subunit rotation decreases. As the nonlinear parameter ε grows, the mechanical step time for the γ subunit rotation drops further. It should be noted that the nonlinearity has no effect on the jumping time from one well to another. The nonlinearity has an effect only on the oscillation time of the γ subunit within the potential well.

2. As the nonlinear parameter ε grows, the relative error of the mechanical step time for the γ subunit rotation between the nonlinear and linear systems grow. At ε = 0.1, the relative error reaches 48.85%. Hence, the nonlinearity of the γ subunit should be considered. Of course, the relative error could not be so large under a condition in which the chemical reaction time is considered simultaneously.

The effects of the system parameters on the rotation rate of the γ subunit are investigated (see Tables 8, 9, 10, 11, 12, 13, 14, 15, and 16). The Tables show:

1. As the concentration [T] of the ATP grows, the rotation rate of the γ subunit grows obviously. For different [T], the rotation rate of the γ subunit grows with increasing nonlinear parameter ε.

For low [T] (0.01 mM), the increase of the rotation rate of the γ subunit with the nonlinear parameter ε is slow. At [T] = 0.01 mM, relative to the rotation rate of the linear system, the rotation rate of the nonlinear system increases 1.75% for ε = 0.25, and 1.80% for ε = 0.5. Here, the concentration of ATP is low and the binding time for ATP to the β subunit is long, which is the rate limiting step which determines the rotation rate of the γ subunit. Hence, the effect of the nonlinearity on the rotation rate of the γ subunit is weak and can be neglected for low ATP concentration.

Table 6.

Mechanical step times for nonlinear system

ε	tnmech (ms)	tfluct30° (ms)	tfluct90° (ms)	trotat30° (ms)	trotat90° (ms)
0.1	0.133	0.031	0.102	0.023	0.047
0.25	0.121	0.028	0.093	0.023	0.047
0.5	0.117	0.027	0.090	0.023	0.047

Table 7.

The error of mechanical step time between nonlinear and linear system

ε	τdrive (pN·nm)	τload (pN·nm)	tmech (ms)	tnmech (ms)	Error (%)
0.1	45	0	0.26	0.133	48.85
0.25	45	0	0.26	0.121	53.46
0.5	45	0	0.26	0.117	55.00

Table 8.

The rotation rates at [T] = 0.01 mM

ε	τdrive (pN·nm)	τload (pN·nm)	[D] (mM)	[P] (mM)	t1/3(ms)	v(rps)
0	45	0	0.3	6	8.120	41.05
0.1	45	0	0.3	6	7.993	41.70
0.25	45	0	0.3	6	7.981	41.77
0.5	45	0	0.3	6	7.977	41.79

Table 9.

The rotation rates at [T] = 0.1 mM

ε	τdrive (pN·nm)	τload (pN·nm)	[D] (mM)	[P] (mM)	t1/3(ms)	v(rps)
0	45	0	0.3	6	3.116	107.0
0.1	45	0	0.3	6	2.989	111.5
0.25	45	0	0.3	6	2.977	112.0
0.5	45	0	0.3	6	2.973	112.1

Table 10.

The rotation rates at [T] = 1 mM

ε	τdrive (pN·nm)	τload (mM)	[D] (mM)	[P] (mM)	t1/3(ms)	v(rps)
0	45	0	0.3	6	2.620	127.2
0.1	45	0	0.3	6	2.493	133.7
0.25	45	0	0.3	6	2.481	134.4
0.5	45	0	0.3	6	2.477	134.6

Table 11.

The rotation rates at τ_load = 5 pN nm

ε	τdrive	[T]	[D]	[P]	t1/3(ms)	v(rps)
	(pN·nm)	(mM)	(mM)	(mM)
0	45	1	0.3	6	2.629	126.8
0.1	45	1	0.3	6	2.502	133.2
0.25	45	1	0.3	6	2.490	133.9
0.5	45	1	0.3	6	2.486	134.1

Table 12.

The rotation rates at τ_load = 20 pN nm

ε	τdrive (pN·nm)	[T] (mM)	[D] (mM)	[P] (mM)	t1/3(ms)	v(rps)
0	45	1	0.3	6	3.490	95.5
0.1	45	1	0.3	6	3.363	99.1
0.25	45	1	0.3	6	3.351	99.5
0.5	45	1	0.3	6	3.347	99.6

Table 13.

The rotation rates at τ_load = 35 pN·nm

ε	τdrive (pN·nm)	[T] (mM)	[D] (mM)	[P] (mM)	t1/3(ms)	v(rps)
0	45	1	0.3	6	10.384	32.10
0.1	45	1	0.3	6	10.257	32.50
0.25	45	1	0.3	6	10.245	32.54
0.5	45	1	0.3	6	10.241	32.55

Table 14.

The rotation rates at l = 5.5 nm

ε	τdrive (pN·nm)	τload (pN·nm)	[T] (mM)	[D] (mM)	[P] (mM)	v(rps)
0	45	0	1	0.3	6	129.5
0.1	45	0	1	0.3	6	129.6
0.25	45	0	1	0.3	6	129.7
0.5	45	0	1	0.3	6	129.8

Table 15.

The rotation rates at l = 7.5 nm

ε	τdrive (pN·nm)	τload (pN·nm)	[T] (mM)	[D] (mM)	[P] (mM)	v(rps)
0	45	0	1	0.3	6	127.2
0.1	45	0	1	0.3	6	127.3
0.25	45	0	1	0.3	6	127.5
0.5	45	0	1	0.3	6	127.7

Table 16.

The rotation rates at l = 5.5 nm

ε	τdrive (pN·nm)	τload (pN·nm)	[T] (mM)	[D] (mM)	[P] (mM)	v(rps)
0	45	0	1	0.3	6	125.1
0.1	45	0	1	0.3	6	125.2
0.25	45	0	1	0.3	6	125.4
0.5	45	0	1	0.3	6	125.6

As the concentration [T] of the ATP grows, the effect of the nonlinearity on the rotation rate of the γ subunit increases. At [T] = 0.1 mM, relative to the rotation rate of the linear system, that of the nonlinear system increases 4.67% for ε = 0.25, and 4.77% for ε = 0.5. At [T] = 1 mM, relative to that of the linear system, the rotation rate of the nonlinear system increases 5.66% for ε = 0.25, and 5.82% for ε = 0.5. Here, the concentration of ATP is relatively high and the binding time for ATP to the β subunit is short. The rotation rate of the γ subunit depends on the chemical step time and the fluctuation time of the γ subunit within the potential well. The nonlinearity mainly influences the fluctuation time of the γ subunit within the potential well. Hence, the effect of the nonlinearity on the rotation rate of the γ subunit is relatively obvious and should be considered for relatively high ATP concentration.

1. As the load torque τ_load grows, the rotation rate of the γ subunit drops significantly. For different load torque τ_load, the rotation rate of the γ subunit grows with increasing nonlinear parameter ε.

For large τ_load (35 pN nm), the increase of the rotation rate of the γ subunit with nonlinear parameter ε is slow. At τ_load = 35 pN·nm, relative to that of the linear system, the rotation rate of the nonlinear system increases 1.37% for ε = 0.25, and 1.40% for ε = 0.5. Here, the load torque is large and the jumping time of the γ subunit from one potential well to another is long, which is the rate limiting step. Hence, the effect of the nonlinearity on the rotation rate of the γ subunit is weak and can be neglected for large load torque.

As the load torque drops, the effect of the nonlinearity on the rotation rate of the γ subunit increases. At τ_load = 20 pN·nm, relative to that of the linear system, the rotation rate of the nonlinear system increases 4.19% for ε = 0.25, and 4.29% for ε = 0.5. At τ_load = 5 pN nm, relative to that of the linear system, the rotation rate of the nonlinear system increases 5.60% for ε = 0.25, and 5.76% for ε = 0.5. Here, the load torque is relatively small and the jumping time of the γ subunit from one potential well to another is short. The rotation rate of the γ subunit depends on the chemical step time, the jumping time of the γ subunit from one potential well to another and the fluctuation time of the γ subunit within potential well. Hence, the effect of the nonlinearity on the rotating rate of the γ subunit is relatively substantial and should be considered for relatively small load torque.

1. As the length l of the γ subunit grows, the rotation rate of the γ subunit drops. For different length l, the rotation rate of the γ subunit grows with increasing nonlinear parameter ε.

For different length l (5.5, 7.5, and 9.5 nm), the increase of the rotation rate of the γ subunit with nonlinear parameter ε is slow. At l = 5.5 nm, relative to that of the linear system, the rotation rate of the nonlinear system increases 0.15% for ε = 0.25, and 0.23% for ε = 0.5.

As the length l grows, the effect of the nonlinearity on the rotation rate of the γ subunit increases slightly. At l = 7.5 nm, relative to that of the linear system, the rotating rate of the nonlinear system increases 0.24% for ε = 0.25, and 0.39% for ε = 0.5. At l = 9.5 nm, the rotating rate of the nonlinear system increases 0.24% for ε = 0.25, and 0.40% for ε = 0.5. Hence, the nonlinear effect of the length l changes on the rotation rate of the γ subunit is small and can be neglected for the different lengths.

In a word, the rotation rate of the γ subunit depends on the chemical step time and the fluctuation time of the γ subunit within the potential well. The nonlinearity mainly influences the fluctuation time of the γ subunit within the potential well. Hence, the effect of the nonlinearity on the rotation rate of the γ subunit is relatively obvious and should be considered for relatively high ATP concentration because the chemical step time is short at high ATP concentration and the effects of the fluctuation time of the γ subunit within potential well on its rotation rate are quite clear.

Conclusions

In this paper, a nonlinear chemo-mechanical coupled dynamic model of the F₁ motor is proposed. Nonlinear vibration frequencies of the γ shaft and the nonlinear stochastic response of the elastic γ shaft to thermal excitation are investigated. The results show:

1. The stiffness nonlinearity of the γ shaft causes an increase of the vibration frequency for the F₁ motor, which increases the motor's rotation rate.

2. When the concentration of ATP is relatively high, the effects of the stiffness nonlinearity on the rotation rates of the F₁ motor are obvious and should be considered.