Optimizing the Efficiency and Power of Molecular Chemical Engines
Abstract Book of the XXIII IUPAP International Conference on Statistical Physics, Genova, Italy, (9-13 July 2007)Abstract
Biological molecular motors are chemical engines which convert
the free energy released by a chemical reaction
into mechanical work. The efficiency $\eta$ of energy
transduction and the power (work done per unit time) $p$ are important
quantities to evaluate the performance of such molecular chemical
engines. However, these quantities are conflicting: high efficiencies
are realized when the engine operates slowly (i.e. with low power),
while the efficiency tends to be low when the machine produces
high power.
An engine working with (moderately) large values of
both efficiency and power may be considered as being of
high performance, and we would like to propose $q = p$ as a
figure of merit to measure the performance of
the molecular chemical engine.
The purpose of the present work is to investigate the conditions for
optimizing the figure of merit $q$.
We work with a simple ratchet-type model for a molecular chemical engine
studied by us K. Sasaki, R. Kanada, and S. Amari, J. Phys. Soc. Jpn.
76 (2007) 023003;
this model takes into account the binding and release of both reactant
and product molecules to and from the engine, and hence enables us to
study the effects of the concentrations $\rho_A$ and $\rho_B$ of
these molecules as well as the load torque $L$ against which
the engine does work.
Once the design of the engine is specified, the set of parameters
$\rho_A$, $\rho_B$, and $L$ that maximizes $q$ is searched.
(The figure shows an example of the dependence of $q$ on
$\rho_A$ and $\rho_B$ as a contour plot for a certain value of $L$.)
This set of parameters provides the optimal condition for a
given design of the engine.
The maximum value, $q_max$, of $q$ depends
on the design of the engine.
It is found that for a well designed engine, i.e. an engine with large $q_max$,
we have $2/3$ and $p (8/9)p_max$ under the condition that $q_max$ is realized,
where $p_max$ is the upper limit of power which the engine of a given design can produce.
The present study also indicates that the F$_1$ motor
(the F$_1$-ATPase) is a well designed engine in this sense,
and that the condition in the cell is likely to be close to optimal one.