204 6.1  Introduction

in Chapters 3 and 4. However, an important point to note here about single-​molecule methods

concerns the ergodic hypothesis of statistical thermodynamics. The ergodic hypothesis

maintains that there is an equivalence between ensemble and single-​molecule properties. In

essence, over long periods of time, all accessible microstates are equally probable. This means

that an ensemble average measurement (e.g., obtained from the mean average from many

thousands of molecules) will be the same as the time-​averaged measurement taken from one

single molecule over a long period of time. The key difference with a single-​molecule experi­

ment is that one can sample the whole probability distribution of all microstates as opposed

to just determining the mean value from all microstates as is the case from a bulk ensemble

average experiment, though the caveat is that in practice this often involves generating sig­

nificant amounts of data from single-​molecule experiments to properly sample the under­

lying probability distribution.

KEY POINT 6.1

The ergodic hypothesis, that all accessible microstates with the same energy are equally

probable over a long time, is relevant to single-​molecule methods since it implies that

the population mean measurement from a bulk ensemble experiment, involving typ­

ically several thousand molecules or more, will be the same as the mean of several

measurements made on a single molecule sampled over a long period of time.

Statistical thermodynamics implicitly assumes ensemble average parameters. That is, a

system with many, many particles. For example, a single microliter of water contains ~10¹⁹

molecules. To apply the same concepts to a single molecule requires the ergodic hypothesis.

Intuitively, one might think that the mean average property of thousands upon thousands

of molecules is an adequate description for any given single molecule. In some very simple,

or exceptional, molecular systems, this is, in fact, the case. However, in general, this is not

strictly true. The reason is that single biomolecules often exist in multiple microstates,

which is in general intrinsically related to their biological function. A microstate here is

essentially a measure of the free energy locked into that molecule, which is a combination

of mainly chemical binding energy, the so-​called enthalpy, and energy associated with how

disordered the molecule is, or entropy. There are many molecules that, for example, exist in

several different spatial conformations; a good illustration of which are molecular machines,

whose theory of translocation is discussed later in Chapter 8. In other words, the prime

reason for studying biology at the level of single molecules is the prevalence of molecular

heterogeneity.

In the case of molecular machines, although there may be one single conformation that

has a lower free energy microstate than the others, and thus is the most stable, several other

shorter-​lived conformations exist that are utilized in different stages of force and motion

generation. The mean ensemble average usually looks similar to the most stable of these

different conformations, but this single average parameter tells us very little of the behavior

of the other shorter lived but functionally essential conformational states. What cannot be

done with bulk ensemble average analysis is to probe such multistate molecular systems. The

power of single-​molecule experiments is that these subpopulations of molecular microstates

can be explored directly and individually. Such subpopulations of states are a vital feature of

the proper functioning of natural molecular machines.

As discussed in Chapter 2, there is a fundamental energetic instability in molecular

machines, which allows them to switch between multiple states as part of their underlying

physiological function. There are, however, many experimental biophysical methods that can

be employed in bulk ensemble investigations to synchronize a molecular population. For

example, these include thermal and chemical jumps such as stopped-​flow reactions, electric

and optical methods to align molecules, as well as freezing and/​or crystallizing a population.

A risk with such approaches is that the normal physiological functioning may be different.

Some biological tissues, for example, muscles and cell membranes, are naturally ordered on a