Letter pubs.acs.org/NanoLett
Optimal Loading of Molecular Bonds Henry Hess* Department of Biomedical Engineering, Columbia University, 1210 Amsterdam Ave., New York, New York 10027, United States ABSTRACT: A corollary of the Bell equation of bond rupture is that the bond transfers a maximal impulse during its lifetime when the applied force equals kBT/ x*. It is proposed that the designs of biological systems converge toward loading bonds with this optimal force to minimize the need for self-healing and to optimize energy transfer.
KEYWORDS: Bell equation, bond rupture, muscle, actin, catch bond, kinesin
T
he study of molecular bonds has revealed that there is no single rupture force, which would be similar to the yield strength of engineering materials.1−3 Instead the lifetime of a bond decreases exponentially from its unloaded lifetime with increasing applied force. A so far unexamined question is at which force a bond should be loaded, given an absence of a defined transition between “safe” and “unsafe” loads. In first approximation, the unbinding rate of a bond under a loading force is described by the Bell equation:4 koff = k 0 × e Fx * /kBT
(1)
where koff is the unbinding rate, k0 is the rate of unbinding in the absence of load, F is the applied force, x* is the distance to the transition state along the reaction coordinate, kB is the Boltzmann constant, and T is the temperature. The average lifetime of the bond under a given constant force τ(F) is simply the inverse of the unbinding rate koff(F), so that the average impulse I transferred across the bond is given by: I(F ) = F × k 0−1 × e−Fx * / kBT
(2)
The transferred impulse depends on the applied force (Figure 1A) and is maximized when F = kBT/x*. I propose that the designs of biological systems converge toward loading bonds with this optimal force to minimize the need for selfhealing and to optimize energy transfer. The evolutionary design process in biology and the rational design process in engineering find optima subject to different constraints. In machine design, a constraint is often the lifetime.5 A part subjected to static or cyclic loads is designed by finding a combination of geometry and material which will reduce the stress to below the stress permissible for that material in consideration of the required lifetime. The resulting design does not necessarily optimize the transferred impulse. A defining feature of biology (and potentially of future self© XXXX American Chemical Society
Figure 1. (A) Impulse transfer across a bond peaks at a force equal to kBT/x*. (B) The replacement rate kr required for structural maintenance as a function of the number of bonds N sharing a load F.
healing materials) is that many mechanically loaded structures are continuously renewed. This introduces an additional degree of freedom into the design process: the replacement rate. Received: August 24, 2012 Revised: September 28, 2012
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dx.doi.org/10.1021/nl303157n | Nano Lett. XXXX, XXX, XXX−XXX
Nano Letters
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Higher loads lead to faster degradation, which can be compensated with faster renewal. If optimal use can be made of each bond, the required turnover activity will be minimized. In the case of N parallel bonds sharing a load F equally, the rate at which bonds need to be replaced due to failure, kr, is given by the product of the number of bonds and the individual failure rate. The replacement rate is minimized for a number of bonds N = Fx*/kBT (Figure 1B), which is when the share of the load for each bond is equal to the optimal force of F* = kBT/x*. Evolution toward optimal impulse transfer likely occurred in cardiac muscle tissue, where well-defined molecular structures are constantly loaded by mechanical forces and renewed.6 The basic organizing unit of striated muscle is the sarcomere, which consists of thin (actin) and thick (myosin) filaments arranged on intercalating hexagonal arrays.7 The mechanical properties of actin filaments have been studied in detail, and the distance to the transition state for the breaking of actin−actin bonds has been measured to be x* = 0.15 nm,8 which implies an optimal force F* of 30 pN. Given the spacing between thin filaments of about 50 nm, the actin filaments are loaded with the optimal force when the muscle experiences a mechanical stress of 30 kPa, which is a value roughly centered in the range of temporally and spatially varying stresses in the heart.9 Of course, the basic argument is complicated by temporally varying stresses. It applies to an “average” force experienced by the actin filament, which is not as readily available as the often tabulated forces under isometric tension or at maximum power for skeletal muscle.10,11 The loading of individual molecular bonds may also evolve toward an optimal force, since, if the bond is used to pull against an opposing load moving at an average velocity v, the mechanical work performed by the bond is maximized for F = kBT/x*, simultaneously with the impulse transfer. For example, the characteristic force F* for a kinesin−tubulin bond is 3 pN and roughly equals the load at which kinesin-1 has its maximal power output.12−14 Similarly, the characteristic forces of 20 pN for the strongly binding conformational state of MannoseFimH or PSGL-1−P-selectin catch bonds are close to the load at which the lifetime of these catch bonds peaks.15,16 The identification of physical constraints for biological evolution has been of long-standing interest not only in biology17 but also in engineering.18 The existence of an optimal load as derived from the Bell equation describing the rupture of molecular bonds is a striking example of such a physical constraint at the nanoscale. The resulting design rule is of use in mechanobiology as well as nanotechnology, where the lifetime of mechanically active nanomachines has to be optimized.
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Letter
REFERENCES
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AUTHOR INFORMATION
Corresponding Author
*Email:
[email protected]. Tel.: (212) 854-7749. Fax: (212) 854-8725. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS I gratefully acknowledge financial support from the National Science Foundation under grant CMMI-0926780 and helpful discussions with Kevin Costa, Emmanuel Dumont, Manu Forero, and Takahiro Nitta. B
dx.doi.org/10.1021/nl303157n | Nano Lett. XXXX, XXX, XXX−XXX
