Mechanisms for achieving high speed and efficiency in biomolecular machines
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Contributed by Ken A. Dill, January 28, 2019 (sent for review July 16, 2018; reviewed by Jonathon Howard and Daniel M. Zuckerman)

Significance
Molecular machines are cellular proteins that transduce energy. They can convert energy to motion, recharge ATP stores by harnessing ion flows, and pump ions energetically uphill. Molecular machines are able to be both fast and efficient at the same time. How do they achieve high speeds at high efficiency? Here we explore mechanisms of evolution or synthetic design that can optimize these properties, such as a conformationally driven mechanical step, which varies the conformational free-energy landscape to mitigate large kinetic barriers, and mechanical substeps, which split one large kinetic barrier into multiple smaller ones.
Abstract
How does a biomolecular machine achieve high speed at high efficiency? We explore optimization principles using a simple two-state dynamical model. With this model, we establish physical principles—such as the optimal way to distribute free-energy changes and barriers across the machine cycle—and connect them to biological mechanisms. We find that a machine can achieve high speed without sacrificing efficiency by varying its conformational free energy to directly link the downhill, chemical energy to the uphill, mechanical work and by splitting a large work step into more numerous, smaller substeps. Experimental evidence suggests that these mechanisms are commonly used by biomolecular machines. This model is useful for exploring questions of evolution and optimization in molecular machines.
We are interested in how biomolecular machines can be optimized for speed and efficiency. Some such machines achieve both speed and efficiency. For example, on an average day a person synthesizes, uses, and recycles his or her body weight in ATP (1, 2). All of this ATP is synthesized by the motor
For a particular machine with a fixed mechanism, going faster comes at the expense of efficiency. However, through evolution, a machine can develop mechanisms that optimize the entire speed–efficiency curve; they increase speed at no cost to efficiency or vice versa. What are these mechanisms that biomolecular machines have evolved to be both fast and efficient?
Previous studies have identified some specific mechanisms that optimize speed, such as a constant torque generation over the angular coordinate of a rotary motor (5), the evolutionary preference for a rotary mechanism in
A Two-State Model of a Molecular Machine
We model how a machine converts the free energy that is put into the system,
The machine cycle is divided into two steps: one nonmechanical (c) and one mechanical (m). The mechanical step (
We use this model to study a broad class of machines, including those with distinct mechanical steps (such as the transport motors kinesin and myosin) and others that have their work output delocalized or “smeared” across the entire machine cycle (such as Na-K ATPase) (14). Although it is simpler than the reaction mechanisms of these real machines, this two-state model captures many essential features of how a machine can best use its input free energy to overcome the kinetic barriers of performing large uphill work.
Expressing Model Mechanisms with Basic Free-Energy Diagrams.
For illustrating mechanisms in this paper, we use diagrams of free-energy–like functions. But first, we describe what our diagrams mean. The observable quantities in Fig. 1 are the forward and reverse rates. We represent these rates in terms of the following free-energy barrier definitions,
Now, if our machines were at equilibrium, we could illustrate their properties using standard free-energy diagrams based on expressions such as
However, here we are interested in steady-state processes, not at equilibrium. But we can capture insights using similar expressions such as
The basic free-energy changes and barrier heights across a machine’s landscape. For this two-state model,
Now, in terms of the rate coefficients above, we can compute the cycle flux (number of full cycles per unit time) as (SI Appendix, section 1)
Defining Parameters for the Free-Energy Transduction Across the Machine Cycle.
The machine landscape in Fig. 2 is defined by the four barrier heights
First, let λ be the fraction of free energy from the input chemical work that is expended within the mechanical step. Thus,
The basic free-energy landscapes of two theoretical machines. The first machine (orange) has most of its basic free energy drop across the nonmechanical step,
Second, let δ represent the location of the transition state along the mechanical step, which dictates how a change in w will affect the forward
Finally, let N define the number of mechanical substeps used by the machine. When
How Machines Can Optimize Speed and Efficiency
To study machine optimization, we seek ways to maximize the speed of the two-state model, at fixed efficiency, with respect to a model parameter θ by solving
Biological machines are impressively fast, given that the mechanical step has a large barrier that arises not only from uphill work but also from large conformational changes [the mechanical step of myosin V, for example, spans 36 nm (18)]. Intuitively, we might expect speed to decrease exponentially with respect to increasing work,
Maximizing Machine Speed by Varying the Basic Free-Energy Drop Across the Mechanical Step.
We first explore how machine speed and efficiency depend on the λ parameter, which determines the drop in basic free energy across the mechanical step. Here, for simplicity, we assume the machine has a single mechanical step (
Here is more intuition about Eq. 9. The first term on the right-hand side of Eq. 9 (
Fig. 3 shows the landscapes of two theoretical machines. The first one (orange,
The speed–efficiency tradeoffs for a machine with
A Conformationally Driven Mechanical Step Optimizes Speed.
Here, we give a molecular interpretation of
We assume here that all of the chemical free energy
To achieve maximum speed, a molecular machine has an optimal change in conformational free energy. Shown is the speed of a theoretical machine with respect to the conformational free-energy change across the mechanical step (
Fig. 5 shows that a machine having a suboptimal conformational change can run orders of magnitude slower. Conformational energy storage modulates the barrier heights across the machine cycle. A machine with small
Machines Are Faster That Have an “Early” Transition State.
Our model also allows for a parameter δ, which can be interpreted as the position of the transition state along the machine’s mechanical step (Eq. 7). It has been shown that machines are faster if they have a small value of δ (7, 8) and we have shown that this conclusion is general (9). That is,
We note that a machine’s power stroke, depending on how it is defined, may correspond to this optimal value of δ (8, 9, 19) or to the optimal value of λ (20, 21). We do not advocate any specific definition here, but by either definition, a power stroke may refer to the release of conformational strain across a machine’s mechanical step.
Speeding up Machines with Mechanical Substeps.
Machine speed can be optimized by breaking the large output step into smaller alternating chemical and mechanical steps, which we call mechanical substeps. A machine that has four 10-kT–sized barriers is faster than a machine with one 40-kT barrier. This principle applies broadly but is best illustrated for a machine where all of the chemical energy is taken up in the nonmechanical step and all of the work is performed in the forward mechanical step (
The mechanical substeps reduce and distribute the free-energy barriers of the machine, as shown in Fig. 6. Fig. 6, Inset shows that, when performing a large amount of work, a machine with more substeps is several orders of magnitude faster than an analogous machine with one large step. Biological machines can leverage the four separate downhill components of ATP hydrolysis (ATP binding, hydrolysis, ADP release, and
Splitting one large work step (black) into two (red) or four (blue) mechanical substeps. ξ is the reaction coordinate. Modifying a molecular machine to have more substeps reduces the barrier height along the mechanical step. Inset shows that, when performing a large amount of work, a machine with several mechanical substeps can be several orders of magnitude faster than a machine with a single mechanical step. Speed is plotted on a log scale.
Similarly,
Other biological machines also use mechanical substeps. In SI Appendix, section 2.C, we use Eq. 8 to calculate
Biological Machines Are Both Fast and Efficient.
Fig. 7 shows that biomolecular machines in vivo tend to have high efficiencies. For many machines, these in vivo values of input free energy
The in vivo efficiencies of molecular machines. Included are Na-K ATPase; SERCA; the proton PPi pump; the plasma membrane proton pump; PMCA; V-ATPase; myosin II; NCX; NCKX; and animal, Escherichia coli, and chloroplast
Fig. 8 shows the speed–efficiency relationships for various biomolecular machines as measured from in vitro experiments. These machines have tradeoff curves that are much more favorable than would occur in a single Arrhenius barrier process, also shown for reference.
The speed–efficiency relationships for six molecular machines. Speed is normalized with respect to the maximum speed of each machine. These machines maintain a high speed even while operating at high efficiency. The black line shows a reference model of a single-barrier, irreversible, Arrhenius process. Data are from refs. 30 and 32⇓⇓⇓–36. Fits to the data are described in SI Appendix, section 4.
We now rationalize some of the differences in the speed–efficiency behaviors shown in Fig. 8. (We do not discuss
Na-K ATPase has the best speed–efficiency tradeoff of the machines in Fig. 8. This pump hydrolyzes ATP as input to export three Na+ ions from the cell and import two K+ ions uphill against their chemiosmotic gradients. This machine has such a good tradeoff because the output work is delocalized across the entire cycle and therefore the rate-limiting work step (the release of one of the Na+ ions) is only weakly dependent on the total work of the machine (26). This pump effectively has many substeps, although we do not report an exact value since it is difficult to estimate the work contribution over the various transitions (the occlusion, transport, and release of each of the five ions).
The cytoskeletal transport motors myosin V, kinesin, and dynein are fast at high efficiency, but they have fewer substeps than Na-K ATPase [
RNAP has the worst speed–efficiency behavior of the machines in Fig. 8. RNAP synthesizes RNA from a template DNA strand (30). As output work, RNAP generates force that melts the DNA template. This output work is physically separated from the machine itself—RNAP breaks hydrogen bonds that are several base pairs upstream from the motor (31). This physical separation may make it difficult or impossible for RNAP to use the speed-optimizing strategies we have described. The motor does not use mechanical substeps and it has a small value of
Conclusions
We have explored the optimization of speed and efficiency of molecular machines using a simple dynamical model. Our model explores parameters of evolution and synthetic design: not just what is, but what could have been. We find that, at high efficiency, a machine’s speed can be increased several orders of magnitude by coupling the downhill chemical input to the uphill work through an intermediate conformational change and by breaking the large kinetic barrier of a single work step into multiple smaller substeps. The benefit of these mechanisms can be seen in the experimentally determined speed–efficiency tradeoffs of biological machines: They allow machines to maintain a high speed even at high efficiency.
These principles are broadly applicable to molecular machines. But one—the optimal value of λ, the basic free-energy drop across the mechanical step—depends on the amount of work that is performed. Some machines operate under fairly consistent loads or forces, but others undoubtedly experience a range of loads. Although we do not explore it here, it is interesting to consider what machine mechanisms may have evolved to optimize performance over such varying conditions.
Acknowledgments
We thank Dean Astumian, Dan Zuckerman, and Joe Howard for comments and suggestions. We are grateful for support from the Laufer Center for Physical and Quantitative Biology and from NIH Grant GM06359217.
Footnotes
- ↵1To whom correspondence should be addressed. Email: dill{at}laufercenter.org.
Author contributions: J.A.W. and K.A.D. designed research; J.A.W. performed research; J.A.W. analyzed data; and J.A.W. and K.A.D. wrote the paper.
Reviewers: J.H., Yale University; and D.M.Z., Oregon Health & Science University.
The authors declare no conflict of interest.
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1812149116/-/DCSupplemental.
Published under the PNAS license.
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