The Power Stroke Mechanism
The Shortening of a (Statistical) Mechanical Spring
"A theory is the more impressive the greater the simplicity of its premises, the more different kinds of things it relates, and the more extended its area of applicability. Therefore the deep impression that classical thermodynamics made upon me. It is the only physical theory of universal content which I am convinced will never be overthrown, within the framework of applicability of its basic concepts." -Albert Einstein
“That a soft jelly should suddenly… change its shape and lift a thousand times its own weight… is little short of miraculous.” -Albert Szent-Gyorgyi
Muscle Power Output
Muscle power output is the work performed by muscle on its surroundings over time — lifting a weight, pumping blood, propelling a runner forward. The mechanism of muscle power output is called the power stroke.
Formally, a power stroke is the shortening of a spring that was previously stretched with force generation (see previous post). But what spring shortens? That depends on the model.
In molecular models, individual molecules are assigned Hookean springs. The shortening of molecular springs is defined as the molecular power stroke.
In macroscopic models, a system spring defines muscle mechanics. The shortening of a system spring is the system power stroke.
If the whole were merely the sum of its parts — if the system spring were just the effective spring of a network of molecular springs — then the two models would describe the same thing only at different scales.
But this is not true of thermodynamic springs (see here). And for the same reason, it’s not true of chemical thermodynamics.
Chemistry Demands a Thermodynamic Model
What we really want to know is the chemistry of the power stroke: what do molecules actually do during muscle shortening?
In molecular power stroke models, changes in molecular states (the shortening of molecular springs) are assumed to cause changes in the system state (the shortening of muscle).
But chemical thermodynamics tells us the opposite: changes in the system state (e.g., changes in temperature, pressure, or the force of a system spring) change the chemistry of an ensemble of motors.
This is not nuance — it is an inversion of agency.
Do molecular states determine system states?
Or does the system state determine molecular states?
Thermodynamics gives a clear answer: the system state determines molecular states.
The state of the system determines the chemistry of molecules, and the heat that flows through the system generates force and power output.
In 1974, T.L. Hill developed a new molecular formalism for this very reason. He understood that Huxley’s 1957 molecular model was incompatible with chemical thermodynamics.
Thermodynamics Is Not Optional
Biological systems are thermodynamic systems. They operate irreversibly, driven by heat flow (e.g., changes in enthalpy) and changes in entropy. It is impossible to understand how life works without understanding this.
Yet many biologists — and even more mathematicians — don’t understand it.
Two short texts should be required reading:
Carnot’s 1824 essay, The Motive Power of Fire. Great scientists do not indulge in the way we like to think about things. They work to constrain the way we think and then work to disprove those constraints. Carnot wrote about the physical constraints of steam engines. Constraints that contributed to the laws of thermodynamics. Constraints that Einstein viewed as pillars of science.
Don’t dismiss Carnot’s caloric theory describing heat as a massless substance that performs work when it flows like a river through a body from high to low temperatures. While heat is only one form of energy, it is contained and flows through bodies like rivers, heat engines, thermal ratchets, chemical reactions, and muscle to perform work.
Feynman’s 1964 freshman lecture on thermal ratchets. A brilliant exposition of the relationship between the mechanics of the components of a thermodynamic system and Carnot’s statement of thermodynamic efficiency.
Neither of these works is long. Neither involves complex math. Neither is difficult to understand. Thermodynamics is, indeed, much easier to comprehend than a black box of molecular mechanisms within which no-one understands what’s happening.
The Myth of the Molecular Power Stroke
In most molecular models, even their authors struggle to explain how molecular springs shorten when muscle shortens. Those who can don’t because they know the molecular power strokes in their simulations are not real. Instead, they rebrand a force generating switch as a molecular power stroke.
Molecular power strokes are not mechanisms. They are beliefs — rhetorical frameworks defended against thermodynamics. Molecular power strokes are “the way we like to think about” how muscle works. Thermodynamics places testable formal constraints on how muscle works.
The aversion to thermodynamics is not epistemological. It is psychological.
It is a refusal to accept that we do not have causal control over molecules. The energetics of the system does.
Agency Belongs to the System
Carnot and Feynman both showed that system function is not caused by molecules. It is caused by the flow of heat through the system. Carnot wrote:
“Heat can evidently be a cause of motion only by virtue of the changes… of form which it produces in bodies.”
Here, “body” means the entire thermodynamic system.
Because all molecular components of a system are subject to the same thermal fluctuations, isolated molecular mechanisms like molecular power strokes are not possible. The body through which heat flows is the system that thermally fluctuates as a whole.
Examples include:
A thermally fluctuating pawl in a thermal ratchet cannot prevent the ratchet wheel from turning backward to thermally equilibrate with the system.
The thermally fluctuating gate of Maxwell’s demon cannot prevent gas molecules from thermally equilibrating with the system.
No thermally fluctuating mechanism can prevent molecular springs from thermally equilibrating with the muscle system.
Because the system fluctuates as a whole, Maxwell’s demon, perpetual motion ratchets, and molecular power strokes are not possible.
With his 1912 analysis of a thermal ratchet, Smoluchowski showed that his equation of motion must be applied to the entire thermodynamic system, not to selected components within that system.
Feynman showed that the ratchet can turn to perform work only if the ratchet wheel is hotter than the pawl. Force and power output are determined by the flow of heat from high to low temperature — not by the mechanics of the components of the system. The spring-loaded pawl — which has a mechanism resembling a molecular “power stroke” — contributes to neither the force generated nor power output by the system.
System mechanics are determined by the flow of heat through the entire system — the ratchet as a whole. Heat flow is not somehow channeled through only those system components that we decide are molecular mechanisms.
To reject this is to reject thermodynamics. And this is precisely what the muscle field has done. The field continues to believe in molecular power strokes — 200 years after Carnot. Nearly 90 years after A.V. Hill proposed a thermodynamic model of muscle contraction. And 25 years after we published “A Thermodynamics Muscle Model and the Chemical Basis for A.V. Hill’s Muscle Equation”.
The Macroscopic Power Stroke Observed
Figure 1 illustrates how an ensemble of force-generating switches responds to the shortening of a system spring — a system power stroke.
Because it is more difficult for a force-generating switch to stretch a high force spring than a low force spring, the probability that switches, M, are bound to actin, A (A·M·D state) relative to not bound (M·D·P state), decreases with increasing system force, F. The specific relationship is described by Boltzmann’s distribution:
PAMD/PMDP ∝ exp(−Fd/kBT),
where d is the average displacement by the throw of a switch. We have directly observed this relationship in skinned muscle fibers (Fig. 2).
According to this relationship, when the system spring shortens and force decreases, switches respond by binding and partially restoring force. This buffers force during a system power stroke and extends the distance shortened.
This is Le Châtelier’s Principle: systems at equilibrium respond to an external change by re-equilibrating in a way that reverses that change.
Force-generating switches rotate — not as the cause — but in response to muscle’s power stroke. These molecular transitions are correlative, not causal. They are caused by an increase in the entropy of the ensemble of switches (see here).
Boltzmann’s probability distribution describes the change in entropy,
—kB·ln(PAMD/PMDP),
with a power stroke. This is, in principle, Boltzmann’s equation, developed to describe the non-equilibrium transport of particles with heat flow. Only here, it describes the non-equilibrium binding of motors with heat flow (i.e., with the enthalpy of binding).
A confluence of different scientific concepts describes this simple binary system. These many pieces need to be put together.
The Implications
This simple system is described as many things at once:
A binding reaction
An ensemble of force generating switches
Non-equilibrium chemical thermodynamics
A Boltzmann distribution of states
Boltzmann’s equation
An entropic spring
A mechanism for irreversible work
This raises many questions about the relationships between these things:
What is the relationship between a change in entropy, kB·ln(PAMD/PMDP), and chemical potential energy, kBT·ln(PAMD/PMDP)? The former alone describes this binding reaction.
Is “mass action” a chemical pull by entropy or a chemical push by chemical activities? The answer to this question determines the arrow of time.
How does entropy affect chemical kinetics? The answer to this question is a derivation of irreversible chemical thermodynamics and kinetics.
How can electrons and protein switches — on scales separated by many orders of magnitude — have the exact same entropy? The answer to this question solves the problem of scaling.
As the components of a system become smaller, at what point does the system stop functioning as a vehicle for work performed by heat transfer? This question led Planck to discover quantum physics, and on larger scales solves Gibbs’ paradox.
What is the relationship between heat transfer in Carnot’s steam engine, Boltzmann’s equation, a flowing river, Feynman’s ratchet, muscle, chemical reactions, and every irreversible process in the universe? Did Carnot at the age of 27 discover the theory of everything?
These questions all center around some of the great mysteries in science. And as a large-scale binary mechanical system, muscle is the simplest possible model system for answering these questions. After working for over 25 years to answer these questions, I’m writing a book (outlined in the Hand and the Arrow) describing what we’ve discovered.
The answers upend molecular determinism and reveal a universe governed not by causal authority, but by the top-down authority of systems within systems defined across many scales. On a given scale, energy flows down entropic wells through the bodies that define those wells to generate emergent system behaviors.
Coming Up
In future posts I walk through the answers to these questions and further explore the implication that molecular determinism is a myth. Numerous mathematical equivalences eerily suggest that every mechanism of system function that appears to be deterministic is correlative. The cause is always an increase in the entropy of the system.
The forms of all bodies — including our own — are randomly and irreversibly pulled into the future by the flow of energy down vast entropic landscapes a priori defined by those bodies. An ensemble of bodies defines new bodies on larger scales with new entropic landscapes and higher-order irreversible function. This is the mechanism of protein folding and self assembly.
Crazy? Follow along. Unlike deterministic mechanisms where the number of mechanisms proposed is unconstrained — i.e., none can be disproven — in thermodynamics the number of possible mechanisms (i.e., entropy) is formally constrained and can be disproven. So, let me know when I make a mistake.