# Capillary muscle

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Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved March 6, 2015 (received for review October 8, 2014)

## Significance

The force generated by a muscle decreases when its contraction velocity increases. This rheology follows the heuristic Hill’s equation. Here, we design and study a capillary analog in which the myosin–actin interaction is replaced by the wetting affinity between a Newtonian silicone oil and a steel rod. This device contracts and generates a force that also decreases with increasing contraction speed, following the same Hill’s equation. The physical origin of this attractive equation is discussed together with the analogy between the actin–myosin system and the capillary one. Apart from its academic interest, this capillary muscle can also have extensions in technology to produce forces and motion using surface chemical energies.

## Abstract

The contraction of a muscle generates a force that decreases when increasing the contraction velocity. This “hyperbolic” force–velocity relationship has been known since the seminal work of A. V. Hill in 1938 [Hill AV (1938) *Proc R Soc Lond B Biol Sci* 126(843):136–195]. Hill’s heuristic equation is still used, and the sliding-filament theory for the sarcomere [Huxley H, Hanson J (1954) *Nature* 173(4412):973–976; Huxley AF, Niedergerke R (1954) *Nature* 173(4412):971–973] suggested how its different parameters can be related to the molecular origin of the force generator [Huxley AF (1957) *Prog Biophys Biophys Chem* 7:255–318; Deshcherevskiĭ VI (1968) *Biofizika* 13(5):928–935]. Here, we develop a capillary analog of the sarcomere obeying Hill’s equation and discuss its analogy with muscles.

From 1487 to 1516, Leonardo da Vinci planned to write a treatise on human anatomy. The book never appeared, but many drawings and writings have been conserved, mainly at the royal collection at Windsor (1):After a demonstration of all of the parts of the limbs of man and other animals you will represent the proper method of action of these limbs, that is, in rising after lying down, in moving, running and jumping in various attitudes, in lifting and carrying heavy weights, in throwing things to a distance and in swimming and in every act you will show which limbs and which muscles are the causes of the said actions and especially in the play of the arms. (2, 3)

Apart from Leonardo’s attempts, the understanding of muscle contraction has been a long quest since antiquity and the work of Hippocrates of Cos (4). The topological structure of muscles was described in the anatomical studies by Andreas Vesalius in 1543 (5) and the static force generated was quantified in the first biomechanics treatise of Giovanni Borelli in 1680 (Fig. 1*A*) (6). One realizes the difficulties associated with the understanding of the force generation mechanism by comparing the scale at which the force is used (typically the body scale:

Despite the complexity of the muscular system, the relation between the force *F* needed to move a given load and the velocity *v* of the motion is accessible via macroscopic experiments such as the one from Wilkie sketched in Fig. 1*B* (11). Here, a constant force *E*, and one records the maximal speed of contraction, *C*. The force reaches its maximum *C* for two subjects (D.W. and L.M. in ref. 11), using the values **1**) is found to apply to almost all muscle types and over various species (13).

The contractile muscular machinery is made of parallel muscle cells that extend from one tendon to another, which connect to bones. A muscle cell is composed of nuclei and myofibrils, a linear assembly of sarcomeres, the elementary contractile unit. The typical size of sarcomeres is 3 μm, so that their number in myofibril of a 30-cm muscle cell is on the order of *C1* and *C2*). When a neuron stimulates a muscle cell, an action potential sweeps over the plasma membrane of the muscle cell. The action potential releases internal stores of calcium that flow through the muscle cell and trigger a contraction (*C2*). Actin and myosin filaments are juxtaposed but cannot interact in the absence of calcium (relaxed-state *C1*). With calcium, the myosin-binding sites are open on the actin filaments, and ATP makes the myosin motors crawl along the actin, resulting in a contraction of the muscle fiber (*C2*) (14, 15). The interaction energy increases with the number of cross-bridges, namely with the surface between actin and myosin threads.

Hill’s equation is a heuristic law and its connection to the sliding-filament model has first been established via adjustable correlations (16) and later via strong theoretical assumptions (17). The purpose of the present article is to build a capillary analog of the sliding-filament model, to record the corresponding force–velocity relationship, and to show how this minimal model system leads to Hill’s equation.

## Capillary Muscle: Setup and Results

Our device is sketched in Fig. 2 *A* and *B*. A cylindrical glass tube (length, *η* = 0.1–1 Pa⋅s; surface tension, *F* (Fig. 2*B*). We denote

An example of a capillary contraction is presented in Fig. 2*D*, where oil has a viscosity *η* = 1 Pa⋅s. As soon as the contact is established between the solid rod and the wetting liquid in the tube (top image), a capillary force *F*_{0} ≈ 100 μN attracts the tube, which moves to the right. The “capillary muscle” contracts and it generates a force as soon as the hook meets the vertical fiber. The time indicated on each picture reveals a nonlinear contraction. We have achieved contractions for two different *B*.

We extract from the deflection *F* decreases as the velocity *v* increases in a way reminiscent of the one observed with muscles (Fig. 1*C*).

## Capillary Muscle: Model

In the above experiment, the capillary driving force is *F* related to the elastic glass fiber, the viscous friction *D*, we get *F*_{0} ≈ 100 μN. In that experiment, the mass of the compound (filled tube plus floater) is *G* for the characteristic acceleration, we evaluate in Fig. 2*D* the inertia term

### Elastic Force, *F*.

The deflection of a thin elastic rod (diameter, *F* is a classical problem since Euler’s elastica (18). In the small slope limit, the elastica predicts a linear relationship between the force *F* and the deflection

where *E* and *I* stand for Young’s modulus and moment of inertia, respectively. For a cylindrical glass fiber, **2** using wetting liquids of low viscosity, to reduce the time to reach equilibrium. Because the applied force *r* of the steel wire (Fig. 2) and on surface tension, we varied both parameters to increase the range of loads. For each experiment, once the contact between the wire and the wetting liquid is established, we wait for equilibrium and measure the final fiber deflection *A*, from which we deduce a stiffness *k* = 3.3 μN/mm. With *r*_{f} = 120 μm and *L*_{f} = 21 cm, the value expected from Eq. **2** is 3.4 μN/mm, in good agreement with the experiment.

### Viscous Force, *F*_{η}.

Without glass fiber, Fig. 4*B* shows that the tube moves along the steel wire with a diffusive type of dynamics: *Inset* in Fig. 4*B*).

As for Washburn’s imbibition (19, 20), the diffusive-like behavior results from a balance between a constant driving force (*α* is a coefficient accounting for the exact structure of the velocity profile in the tube (*η* = 1 Pa⋅s, we get from the fit (thin solid line) in Fig. 4*B*:

### Hill’s Equation for a Capillary Muscle.

The quasi-steady equilibrium **1**). The solid lines in Fig. 3 correspond to hyperbolic fits obtained with Eq. **4** and *k* have not been changed,

## The Sliding-Filament Model and Its Capillary Analog

### Capillary Analog.

Analogies between liquids and tissues have led to valuable findings in the context of embryonic mutual envelopment (21) or tissue spreading (22). Steinberg showed that embryonic tissues may behave like liquids and can be characterized by a well-defined surface tension (23). This analogy lately extended to the spreading dynamics of cellular aggregates (24, 25). Here, we pursue this kind of analogy to illustrate the myosin–actin interaction in a muscle. The muscle contraction results from the actin–myosin interactions. Each myosin head can be in two different states, attached to the actin filament or detached from the actin filament. We denote *l* is the typical size of the myosin head. This implies an equivalent surface energy *S* is the attachable surface area). Because −

Our capillary device can be seen as a physical analog of the myosin/actin system, as sketched in Fig. 2: the solid rod (“myosin”) slides in a wetting liquid (“actin”). As the myosin rod penetrates the actin tube by a distance *θ* is the contact angle and *γ* is the liquid–vapor surface tension (26). The wetting limit (**5**. The liquid viscosity leads to dissipation and it mimics the energy consumption of the muscle, which we now discuss together with its link with Hill’s equation.

### Sliding Filament and Hill’s Equation.

According to Needham (4), “the first hints of the sliding-filament mechanism of contraction were given by the low-angle X-ray diffraction patterns obtained by H. E. Huxley with living and glycerol-extracted muscle” (27). A theory for the contraction based on this sliding-filament model was then proposed by A. F. Huxley in 1957 (16), where different parameters were chosen to approach Hill’s equation. In 1968, Deshcherevskii (17) proposed to derive Hill’s equation using some assumptions on the sliding-filament model. We present the main steps of this model and then establish the connection with capillary muscles.

In the sliding-filament model, the force is generated by myosin heads connecting myosin to thin actin filaments (Fig. 2*C*). This scenario has been confirmed since refs. 28 and 29, and we summarize the force cycle in Fig. 5*A*. In the absence of ATP, myosin heads are attached to actin. Although this state is very short in living muscle, it is responsible for muscle stiffness in death. As binding ATP, myosin heads release from actin filaments, which requires energy.

Deshcherevskii considered three main stages for the force cycle (Fig. 5*B*). A myosin head is either “free” (A), or developing an active force (B), or detaching (with a breaking force) (C). Each head moves from one state to another one following the sequence A–B–C–A. Denoting *n* and *m* as the number of myosin heads in the states B and C and *l* is the mean value of conformational transformation of the myosin head during the power stroke (*v* is the velocity of relative displacement of the threads, the ratio *v* is a constant, the populations of the three stages A, B, and C remain constant on average and one deduces *f*, found to be on the order of *n* and *m*, we directly get Eq. **1** with

Identification with Eq. **4** provides some insight about the analogy between sliding filaments and capillary muscles. As expected from the previous section, the maximal force **4** corresponds to the force **4** is found here to be only a function of the reaction rates

### Inverted Capillary Motion.

Our capillary device was found to generate a contractile force analogous to that of a real muscle. From a practical point of view, it is worth exploring the possibility of inducing a reverse motion. To do so, we inverted the wettability by treating the steel wire with a hydrophobic colloidal suspension (Glaco; Soft99). After drying the solvent, the fiber was observed to be superhydrophobic (advancing and receding angles of *D*) to minimize the contact between the liquid and the wire. This behavior is reminiscent of the transition from contraction to relaxation (*C2–C1* transition in Fig. 2). If the device is let free, the wire gets eventually expelled from the tube. This experiment also shows that a system of tunable wettability [by means of temperature or light (34)] should be able to successively generate contraction and relaxation.

## Conclusion and Perspectives

We have designed a minimal capillary model of force generator following Hill’s contraction law. The model is based on three main characteristics, which exist both in muscles and in its capillary analog:

*i*) Inertia is not involved and the system contracts in a quasi-steady regime.*ii*) The force driving the contraction is based on surface affinity between two surfaces that can slide with respect to each other.*iii*) The device not only generates a force but also dissipates energy.

It was often proposed, in the biomechanics community, that Hill’s equation can be recovered with a spring-dashpot macroscopic model involving a non-Newtonian dissipation (13). Here, we showed that a Newtonian fluid also allows to recover the hyperbolic force–velocity relation, provided one accounts for the sliding structure of the contraction, which induces a nonlinear term “*xv*” in the dissipation, which is necessary to get Hill’s equation. The analogy might be pursued to understand other systems, such as living single cells, which have been lately found to also follow Hill’s equation (35).

Finally, a capillary muscle can be discussed in terms of innovation. We have shown that tuning the contact angle allows the cell either to contract or to relax. As in real muscles, this elementary contractile unit can be coupled to other identical contractile units, in series or parallel to increase the contraction speed or the force generated. Microfluidics and robotics are possible areas of application.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: clanet{at}ladhyx.polytechnique.fr.

Author contributions: C. Cohen, T.M., D.Q., and C. Clanet designed research, performed research, analyzed data, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

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