Part II. Application of Statistical Mechanics in Biological Phenomena^6.1

Here, in Part II, which is the second half of this book, I will explain how the statistical mechanics concepts described in Part I can be related to biological phenomena, by employing an approach based on the dice and chips game. Using the technique of observation and measurement of the movement of single molecules via a microscope, I will focus on specific examples of single molecule behaviors. Chapter 6 covers topics related to the viscosities of liquids and gases. I will demonstrate that the movement of objects in gases and liquids, and chemical reactions can be understood based on the dice and chips game.

Chapter 7 addresses the concept of local temperature at a molecular level—a concept that I proposed^6.2. Chapter 8 examines the topic of Brownian motion, elucidated by Einstein by invoking the existence of molecules. If the readers find Chapters 6 and 7 difficult to understand, I may kindly be excused, as they are quite specialized. In Chapter 8, I hope you will have fun, as you will roll a dice and write the outcome. In all parts of the Part II, I will also discuss how two new and interesting problems, one in physics called “statistical mechanics of a small number of molecules” and the other in biology called “role of molecular fluctuations (looseness) inside a living organism” come together to produce a fascinating phenomenon.

Chapter 6: Energy Exchange and Required Time

As described in Chapter 5, a person’s experience of becoming rich is gained at the third stage (Table 5.1, 3: Each person experiences the best). It is worth considering how long it would take to achieve this outcome. This question is of great significance in chemical reaction theory.

6-1: Motion in gases (low viscosity)^6.3

Consider a sphere (which can be a macrosphere) suspended in a gas composed of small molecules. The sphere can exchange energy with the surrounding gas molecules via collisions. Let us now consider how much energy the sphere receives when it collides with the gas molecules, and how long it takes for the sphere to accumulate a large amount of energy by chance. As previously shown in Chapter 2 (Table 2.7), the number of chips distributed to all the boxes will eventually become a stable time-independent exponential distribution. As the distribution becomes exponential because of the others (i.e., the environment)^6.4, it is inconsequential even if the box of interest is an especially large box. Therefore, the molecule of interest can be macrospheres, different in size from other gas molecules.

The collisions of the surrounding molecules with the sphere will accelerate the latter in one particular direction, while, collisions with some other molecules in the opposite direction will tend to suppress its motion. Thus, the collisions cause both acceleration and deceleration of the motion of the sphere—a phenomenon related to time-reversal symmetry. While it is difficult to image the collisions that accelerate the sphere, it is much easier to consider collisions that decelerate the sphere by bringing in the concept of friction. I am not saying that this friction-based model is accurate or appropriate; however, I think it provides an easier option to explain the phenomenon. Considering the deceleration of motion from the viewpoint of friction or “viscosity”^6.5, it can be surmised that the speed of the large sphere is reduced on account of frictional force. Moving the sphere suddenly with a large kinetic energy gives rise to the question—“how long does it take for this kinetic energy to decay due to the frictional force of the gaseous molecules”? We can calculate the energy loss and, by thinking reverse, estimate the time it takes to accumulate a certain amount of energy from the gas molecular collisions. The decay would be faster if the gaseous friction is higher; in other words, employing the concept of time reversal, we can imagine that the higher the viscosity of the gas, the faster our example macrosphere would receive energy, considering the fact that there would be more chances of collisions. Therefore, if a sphere becomes chemically reactive after receiving a certain amount of energy, we can assume that the reaction rate increases when the “viscosity” of the surrounding gas increases. If the friction is high, the speed of the sphere decreases rapidly. Following the principle of time reversal, the larger the friction, the faster the energy can be extracted. The key aspect here is the reversal of time. I would like you to become familiar with and get used to this concept, which you may find very useful. The two concepts, the time-reversal symmetry and “now” is always the peak moment followed by decays towards the average value (Chapter 4), can be used as I described above.

Employing the time reversal concept, consider the following case: In the dice and chips game that I was playing, while exchanging the chips, by chance, my chips kept increasing in number. After a sequence of increases, this number started to decrease. If we were to plot the number of chips in my box versus time, such a plot would likely exhibit many peaks and troughs. However, if you were to flip and examine the plot, it would appear more or less the same as it did before. This is indicated in Fig. 6.1 A and B. Now, as an example of a non-time reversal case, let us also draw a sawtooth wave pattern (Fig. 6.1D). The plot-line ascends slowly, but descends faster; thus, when it is flipped over, a clear difference can be observed. Now, the ascent would be fast, while the descent would be slow. Hence, we understand that the patterns are obviously different with time reversal. This is the scenario, wherein there is a time asymmetry. In clear contrast, we did not notice any asymmetry with the chips and rolling dice scenario (Fig. 6.1 A, B). Although it may sometimes happen that going up is faster than going down and vice versa, these happen with the same frequency. When we examine the long-term plots, both forward-in-time and reversal-in-time appear to be almost the same. Thus, we state that the patterns appear to be the same in time reversal.

Figure 6.1 

Time-reversal symmetry. A: The increase or decrease in the number of chips of the two persons in the simulation that six-person exchange 30 chips for 10,000 times (the first person has black chips, while the second person has gray chips). The rules of the game are same as those of Exercise 1. The horizontal axis represents the number of exchanges and the vertical axis represents the number of chips held by each person. B: The right and left of A are reversed. C: Enlarged view of A (from the 3,700^th to 4,500^th exchange). D: Example of time reversal asymmetry. Reversing the time of slow ascent and fast descent (left) results in a fast ascent and slow descent (right). The arrows indicate the direction of progression of time and the vertical axis represents the number of chips.

Coming back to our sphere surrounded by gas molecules, any rapid movement of the sphere will be quickly dampened, depending on the amount of friction. However, because of the time reversal principle, any stationary sphere can once again quickly gain velocity from numerous collisions with the surrounding molecules; that is, the sphere would gain velocity faster when the density (and hence friction) is greater. Thus, we can say that owing to the time reversal symmetry concept, the sphere can both quickly gain speed and equally quickly lose speed.

By all means, please roll the dice and experience this for yourself. Even if you do it once, you will remember it for the rest of your life, and it will be a useful knowledge. If you just listen to my talk and do not perform the experiment, you will not remember anything.

6-2: Motion in liquids (high viscosity)^6.6

The discussion, thus far has been based on the motion in a gas (low viscosity or friction); however, this phenomenon is different in a liquid (typically higher viscosities than gases). Let us consider the vibrational (bending) motion of a cylindrical object in a liquid. For example, consider the bending of a fibrous protein polymer F-actin. Fig. 6.2 shows a diagram depicting the bending of F-actin in a liquid (with the bending motion driven by a thermal fluctuation). When molecules of the surrounding liquid, which have thermal motion, collide with the thin F-actin filaments, the latter are subjected to bending. If the frictional force is significant, the bent F-actin cannot easily return to its original (straight) condition. The higher the liquid viscosity, the longer it takes for the F-actin to bend and return to its straight position (called relaxation time). The relationship between the relaxation time and friction or viscosity in liquids is opposite to that in gases. When we reverse the time, we can see that molecules within a higher-viscosity environment cannot easily bend or receive energy. Therefore, achieving a state of high elastic energy in a liquid environment is difficult to realize. If the system receives elastic energy and uses it for some other reaction, then the higher the viscosity, the less likely that the reaction would occur. This relationship between the viscosity and speed of achieving high-energy states is the opposite of that observed in in a gas.

Figure 6.2 

Bending motion of F-actin in a liquid. The length of F-actin (with heavy meromyosin) is approximately 10 μm. The time interval between adjacent images in the vertical direction is 1/12 s. The relaxation time τ can be expressed as τ ∝ ζ/ε, where ζ is the viscosity (or friction) and ε is the elastic modulus. This information is reprinted from Publication [6.1] Copyright 2023, with permission from Elsevier.

As described in the previous section, when the object of interest (macrosphere) receives energy^6.7 from gas molecules, the higher the friction or viscosity, the greater the probability of the energy being received. However, when the object of interest (F-actin) receives energy^6.8 from liquid molecules, as described in this section, the higher the viscosity, the lower the probability of receiving the energy. In other words, the relationships between the energy received by an object, and the friction/viscosity force on it in a gas and a liquid are opposite in nature^6.9. This is because the densities of the surrounding molecules that collide with the object are different; therefore, the frequencies of the collisions are different. In the case of an object in a gas, because of the lower density of the colliding molecules, the object receives more energy at a higher viscosity (more collisions). However, for an object in a liquid, because the frequency of collisions is basically very high, an object cannot maintain its high energy state. Therefore, the object receives more energy at a lower viscosity. This is the key difference between the two cases.

Nevertheless, when considering the probability of achieving a high-energy state from an equilibrium state, we can reverse this idea and consider the time required for the decay to an average value when placed in a high-energy state. Based on the concept of time reversal symmetry, one can estimate how long it takes for the energy to increase owing to the interaction with the surroundings. This is very important.

6-3: Low and high viscosities: Cis/trans transition of molecules

In the following discussion, I apply the same general concepts as developed in the previous sections (although I am not sure whether they are completely acceptable!). Stilbene is a good example of a chemical that shows different reaction kinetics in the low- and high-viscosity regimes. An exposure to light initiates a transition between trans- and cis-stilbene, with the reaction rate changing with viscosity (Fig. 6.3). In the low-viscosity region, the reaction rate increases as the viscosity increases, similar to the case of gases (Section 6-1). On the contrary, in the high-viscosity region, the reaction rate gradually decreases as the viscosity increases, which is similar to that noticed in liquids (Section 6-2). This change in kinetics was predicted by Kramers^6.10 in 1940, and experimental results were first obtained in the 1980s [6.2–6.4].

Figure 6.3 

Viscosity dependence of cis/trans isomerization rate of stilbene. A: Cis/trans isomerization of stilbene by light irradiation. Stilbene, a hydrocarbon, is isomerized from the trans form (left) to the cis form (right), and vice versa by irradiation using ultraviolet light. B: Isomerization rate (y-axis) vs. viscosity (x-axis). In a low viscosity region (for example, in gases), the reaction rate increases with viscosity, whereas in a high viscosity region (for example, in liquids), the reaction rate decreases as viscosity increases. Diagram created based on [6.3,6.4].

Although it is the job of scientists to develop solid theories and derive dynamics equations applicable to both high and low viscosities, I have introduced them as an analogy so that they can be intuitively understood. A review article on this topic can be found in the Proceedings of the Physical Society of Japan (Butsuri) [6.5]. Although this review provides a more detailed explanation, its major observations are as follows: Molecules in a low-viscosity medium receive collision energy, as in gases (Section 6-1); however, molecules in a high-viscosity medium undergo significant attenuation of energy owing to friction, as in liquids (Section 6-2). Thus, there must be an optimal state somewhere in the middle-viscosity region, for which the energy uptake by the molecules is optimal. This means that the molecules cannot receive energy when the density is too high or too low, but can do so when the density is moderate. If there are too many collisions, energy cannot be stored throughout. If collisions are too sparse, energy cannot be received. Hence, the molecules can receive energy at the highest rate when the collisions are optimal.

6-4: Single molecule measurement of the hydrolysis of ATP

In certain types of chemical reactions, known as first-order reactions, the reaction rate (written as k) is constant, and is related to the reciprocal of the time taken for one molecule to undergo the reaction; therefore, it has the unit of s^–1. If there are n molecules undergoing such a first-order reaction, the number of molecules that react per unit time will be k×n. All the molecules in the pre-reaction state will have equal reaction probabilities and will proceed with the reaction regardless of their prior experience. Reacting with equal probability every moment means that the probabilities of obtaining high energy by chance are equal at any moment. If it gradually accumulates energy and reacts after crossing a certain point, accumulation history will remain even after the stored energy is discharged, but that is not the case. In the time scale we assume here, past history and experience are not considered, and the energy will suddenly become high with the same probability every moment (Table 6.1).

Table 6.1  In fluctuations, an occasional large deviation from an average is a major event

Occasionally, a large deviation from the average occurs in a fluctuation.

At what time interval?

How long does it take to deviate?

Example:

Sphere in a gas: Occasionally attains a high speed.

Twisting, bending: Occasionally causes a large deformation.

Chemical reaction: Occasionally a high energy state and a reaction occur.

In living organisms?

In 1995, an important experiment was conducted to observe individual reactions [6.8]^6.11. Myosin, a molecule that induces muscle movement, hydrolyzes ATP to generate energy^6.12. To directly observe the process, by which myosin would decompose ATP, a fluorescent dye was covalently attached to the ATP molecule. The fluorescent ATP was then added to myosin, fixed on a glass surface, and the state of bond dissociation was observed under a microscope. Using a special instrument called a total internal reflection (evanescent) fluorescence microscope, a bright spot could only be observed when a fluorescent ATP molecule was bound to myosin near the glass surface. However, it was not visible when it moved freely with Brownian motion in the solution without being associated with myosin. When the fluorescent ATP was hydrolyzed to produce fluorescent ADP and phosphoric acid within myosin, the fluorescent ADP dissociated from myosin, and the bright spot disappeared. By examining the frequency distribution (n) over the flickering time t obtained for many such reactions, the exponential distribution shown given by Eq. 6.1 was obtained.

  

$${n\propto e}^{-kt}$$(Eq. 6.1)

Here, k, the reaction rate constant of ATP hydrolysis, facilitated by myosin is approximately 0.05 s^–1.

The fact that the reaction time of one ATP molecule on a myosin molecule is exponentially distributed implies that regardless of the duration that each molecule has been there, the next reaction will occur in a predictable (probabilistic) fashion. For myosin-catalyzed ATP hydrolysis, the rate-determining step (the elementary process with the longest reaction time) involves the separation of phosphate (P). Almost immediately after the ATP molecule binds to myosin, it decomposes into ADP and P; however, does not easily proceed to the next step while still being attached to the myosin molecule. If myosin is isolated, P would finally get separated after a few seconds^6.13. After P is separated, ADP is separated as well. However, to reach the point where P is about to separate, the energy released from the phosphodiester bond must be stored elsewhere in the myosin molecule; in fact, when ATP hydrolysis within a single myosin molecule is examined, energy storage is observed. Hence, I believe that this experiment is a significant achievement.

The experimental results indicate that the reaction exhibits an exponential distribution, similar to that suggested by Eq. 6.1, which means that k can be defined using a one-step reaction of myosin (M) with one ATP molecule, as shown in Eq. 6.2.

  

$$M+ATP \stackrel{k}{\to }M+ADP+P$$(Eq. 6.2)

Comparing Eq.6.2 to the dice and chips game, introduced in Part I, we note that only a small number of boxes (read molecules) will have enough chips (read energy) to undergo the reaction. It is extremely important to remember that occasional reactions occur with the same probability every moment, regardless of the past experience. This guarantees that the chemical reactions can be described by simply using k.

Interestingly, we were able to predict this general behavior using only a dice and four boxes (see Part I). However, a game with four chips in four boxes is insufficient to produce a sudden accumulation of energy (chips) (as opposed to a slow accumulation). It would be better if a few more chips and boxes were used.

In the case of bending of F-actin filaments (described in Section 6-2) the accumulation of large amounts of energy will be different (Fig. 6.4). Here, sufficient energy accumulation does not occur suddenly as in the case of the myosin reaction. However, constant violent collisions of water molecules occur around F-actin, which eventually cause F-actin to start bending. As discussed in Section 6-2, the higher the viscosity of the medium, the longer it takes for F-actin to bend; this situation is different from the rapid accumulation of energy in the myosin-governed reactions. As this is a very important topic, we will return to this in the next section (Section 6-5).

Figure 6.4 

Schematic diagram of the bending motion of F-actin thin filaments in a high-viscosity liquid.

In Section 6-1, we discussed the gain in the energy of the molecules when impacted. To continue that discussion, let us consider the classic story of Rayleigh^6.14, which was originally conceived more than 100 years ago. Here, as shown in Fig. 6.5, is a piston-like object, in which a round plate is placed in a one-dimensional cylinder such that there are no gaps between the plate and the cylinder wall, and in which molecules move while being impacted from both the left and right. If the mass of the disk-shaped object in the middle is M and its velocity is V, then its kinetic energy can be calculated using Eq. 6.3.

Figure 6.5 

Motion of a large cylinder in a gas (Rayleigh’s piston). When gas molecules (black dots) hit the piston (black plate in the middle) from both sides, the piston moves left and right. Both the motion of the plate and breaking of the motion are due to molecular collisions (Brownian motion with a restoring force).

  

$$\epsilon =\frac{1}{2}M{V}^2$$(Eq. 6.3)

At the same time, the average kinetic energy of the impacting molecules is given by Eq. 6.4.

  

$$⟨\frac{1}{2}m{v}^2⟩=\frac{1}{2}{k}_{B}T$$(Eq. 6.4)

where m is the mass of the surrounding molecules; v is their velocity; k_B is the Boltzmann constant; and T is the absolute temperature. By solving the mechanics, Rayleigh proved that the probability distribution ψ for the energy in Eq. 6.1 is given by Eq. 6.5^6.15.

  

$$\psi \propto e^{-\frac{1}{2}M{V}^2/k_{B}T}$$(Eq. 6.5)

However, this is a very classic problem. In a famous book titled “Probability Theory and Statistics” [6.9] by Kodi Husimi^6.16, which was published before World War II (and also reprinted after the war), a description of Rayleigh’s piston, presented in the preceding section, is outlined. Part I of this book describes the formula for the number of distribution methods in a large system. Interestingly, in the later chapters, a popular example of what we now know as ‘chaos’, called “making apple pies” (pie kneading experiment) is described (See the addendum “Coffee break” at the end of this chapter).

6-5: What is viscosity?

In Sections 6-1 and 6-2, we discussed the viscosity property of gases and liquids, and the diverse effects of viscosity on the accumulation of elastic and kinetic energy. In the late 1940s, several researchers attempted to describe viscosity of liquids and gases. As this area has received little attention since then, I would like to reintroduce it.

Shortly after World War II, from 1946 to 1949, Kirkwood^6.17 published a series of papers on how viscosity could be expressed by intermolecular forces [6.10–6.12]^6.18. As viscosity implicitly signifies flow, it refers to a non-equilibrium state (Fig. 6.6). From the molecular radial distribution function (which shows the arrangement of molecules), such a non-equilibrium state can be expressed using Eqs. 6.6 and 6.7.

Figure 6.6 

Deviation and viscosity owing to the flow of molecular distribution. Representation of viscous molecules by Kirkwood.

  

$$\eta =\frac{N^2}{V^2}\frac{1}{15}{\int }_0^{\infty }R\bullet \left(\frac{d\Psi }{dR}\right)\bullet g^{*}\left(R\right)\bullet 2\pi R^{2}dR\text{.}$$(Eq. 6.6)

  

$$g\left(R\right)=g_0\left(R\right)+g^{*}\left(R\right)\bullet \left[flow\right]$$(Eq. 6.7)

where g(R) is the molecular radial distribution function and ψ(R) is the intermolecular force potential.

Part 1 of the series of papers was published in 1946, when he could finally write a thesis after the war, and contains a lengthy preface by Kirkwood [6.10]. This was probably his research during the war; therefore, he wrote it with deep emotions, while thinking about what he was planning to do from then on. As the papers written during this time period were typically of high quality, I recommend that you read them if you have the time. Viscosity is a very basic issue concerning the imparting or receiving of energy through a gas or liquid. Please study this concept and conduct some research. Although it gained significant attention at that time, I suppose that this is not a popular subject now; hence, it is not cited frequently in textbooks. However, the nature of viscosity is worth considering, for example, for estimating the friction when the bacterial flagellar motor is spinning and its relation to the intermolecular forces involved.

6-6: Torsional motion of a mirror suspended in gas

Next, I will discuss the deformation of an object caused by molecules when they impact an object. We have already discussed the Rayleigh’s piston earlier with respect to kinetic energy; here, we discuss torsional energy. To understand how elastic energy is stored, we examine the torsional motion of a mirror suspended in a gas by a thin thread (Fig. 6.7). The torsional motion of this mirror was solved in the collection of problems called “Statistical Mechanics” [6.13] (Chapter 6, Exercise A-6). The torsional energy is given by:

Figure 6.7 

Kappler’s experiment: Torsional motion of a mirror suspended using a thin thread in a gas. Here, the torsional energy is $\epsilon =\frac{1}{2}K{\theta }^2$ (Eq. 6.8), and the energy distribution is proportional to $\text{exp}\left(\frac{-\frac{1}{2}K{\theta }^2}{k_{B}T}\right)$ (Eq. 6.9).

  

$$\epsilon =\frac{1}{2}K{\theta }^2\text{.}$$(Eq. 6.8)

Here, K is the torsional elastic constant and θ is the torsion angle. The energy distribution ψ then becomes:

  

$$\psi \propto \text{exp}\left(\frac{-\frac{1}{2}K{\theta }^2}{k_{B}T}\right)\text{.}$$(Eq. 6.9)

A famous experiment was conducted by Kappler^6.19 in 1931 on the twisting motion of a mirror [6.14]^6.20. A small mirror with a surface area of 1 mm² was suspended in air and irradiated with light. Slight fluctuations in the movement of the mirror were magnified and measured as reflected light. Of course, we will naturally think about these aspects when we perform this type of an experiment. What happens if we change the density of air; in other words, what happens when the density is high or low?

Given that the mirror suspended by a thread has elastic energy, it will swing by itself. Because there is a restoring force due to the elastic force, it will move as shown in Fig. 6.8. As the molecules impact the mirror, when the density of a gas is high, that is, when there are more number of gas molecules, the motion could be as shown in Fig. 6.8B, and it is difficult to restore the mirror against extreme fluctuations. However, in Fig. 6.8D, when the gas density is low, as the molecules only rarely impact the mirror, the mirror oscillates and creates its own elastic motion. Fig. 6.8C shows an intermediate condition. The temperature is the same in all the cases. I once asked the students to solve this problem in a graduate school entrance examination. I presented one of these figures and asked the students to draw a figure that corresponds to a higher pressure and higher temperature. I think it is a pretty good problem. The crux of the problem is not a matter of theory, but of intuition.

Figure 6.8 

Diagram of Kappler’s experiment for optical measurement of the torsional motion of a mirror suspended using a thin thread in a gas. A: Measurement system. B: Fluctuations at 1 atm pressure. C: Fluctuations at a low pressure. D: Fluctuations at an extremely low pressure.

Therefore, this energy distribution becomes an exponential distribution, as represented by Eq. 6.9, even when solving the dynamic equations. Using the average kinetic energy of the gas, the following equation can be obtained:

  

$$⟨{\theta }^2⟩=\frac{k_{B}T}{K}\text{.}$$(Eq. 6.10)

The average fluctuation is determined to be $\frac{1}{2}{k}_{B}T$. If the average value of the kinetic energy of the impacting gas ($\epsilon =⟨\frac{1}{2}m{v}^2⟩$) is set to $\frac{1}{2}{k}_{B}T$, it can be shown that the average value of the elastic energy ($\epsilon =\frac{1}{2}K⟨{\theta }^2⟩)$) of the gas will also be equal to $\frac{1}{2}{k}_{B}T$ (Table 6.2). The original definition of the torsional elastic modulus K is the amount of twist that occurs when an external twisting force F is applied; K is inversely proportional to the twisting angle θ.

Table 6.2  Spontaneous fluctuation of twist angle and ease of twisting when an external force is applied

$⟨{\theta }^2⟩=k_{B}T\frac{1}{K}$

$\left[\begin{array}{c}\text{Mean-square of}\\ \text{the spontaneous fluctuation of}\\ \text{the twist angle}\end{array}\right]\propto T\times \left[\begin{array}{c}\text{ease of twisting}\\ \text{when an external force is applied}\end{array}\right]$

  

$$\frac{\theta }{F}=\frac{1}{K}$$(Eq. 6.11)

If K is large, the object is stiff and difficult to twist. Therefore, if the torsion angle is large when an external force is applied, the viscosity would be low. The elastic modulus can be measured based on this concept. The same elastic modulus K appears in Eq. 6.10 as the coefficient of fluctuation.

Thus, we understand that the elastic modulus, calculated as the recovery force that comes into play when an external force is applied, is same as that calculated from the bending while in thermal equilibrium with the surroundings under a thermal motion (Table 6.3). It has been proven that the elastic modulus, which determines the spontaneous fluctuations, and the elastic modulus of bending when an external force is applied are the same. This implies that the different aspects of statistical mechanics are consistent each other.

Table 6.3  Fluctuation (internal force) and sensitivity (response to the external force)

The friction (the ease of movement) experienced when a movement occurs via an exchange of thermal energy with the surroundings is the same as that experienced due to a movement induced by an external force.

The flexibility (stiffness) exhibited when bending occurs due to an exchange of thermal energy with the surroundings is the same as that exhibited when bending occurs due to an external force.

As the average of the thermal energy is fixed (assuming the temperature is held constant), fluctuation (internal force) [Symbol F020]∝ sensitivity (response, external force)

During the bending motion of F-actin, described in Section 6-2 (Figs. 6.3 and 6.5), as it is already in a liquid, the mean square of the bending thermal motion amplitude will be

  

$$⟨q^2⟩ =2\left(\frac{k_{b}T}{\epsilon }\right)\bullet \frac{L^4}{{\pi }^4}\text{.}$$(Eq. 6.12)

Here, the bending amplitude q depends on the filament length L and ε is the flexural modulus; then the relaxation time (restoration time) τ becomes

  

$$\tau =\left(\frac{\zeta }{\epsilon }\right)\bullet \frac{L^4}{{\pi }^4}\text{.}$$(Eq. 6.13)

In this case, the elastic modulus ε is in the denominator while the viscosity ζ, which is the coefficient of friction, is in the numerator. The dependency is such that when the viscosity is high, the recovery time will increase. This is a very effective measure, as the length term is raised to the 4^th power.

As we have noticed, the restoration time for the bending caused by an external force can be used to estimate the friction during a bending thermal motion. In addition, the elastic modulus during bending serves as a measure of the material resilience. The fact that the coefficient of friction and elastic modulus estimated during the bending thermal motion caused by an external force are consistent provides confidence that these general physical calculation methods are valid.

To reiterate, the friction (force-resisting movement) calculated from the molecular movement caused by the exchange of thermal energy with the surroundings is the same as the friction calculated from the movement induced by an external force. It is important to understand that the flexibility observed during bending caused by the exchange of thermal energy with the surroundings is the same as that observed during the bending owing to an external force. For a moment, we may wonder why this happens; but, this is how it all works.

I think the first proper treatment of these aspects was performed in a study of Brownian motion by Einstein^6.21 in a paper in 1905 (presented in detail in Section 7-7 and Chapter 8). In this study, Einstein proved that the frictional coefficient that appears during a Brownian motion and that of macroscopic friction have the same meaning [6.15]. Although I am not sure if this is completely true, it is a principle that is related to statistical mechanics.

In summary, by measuring the fluctuations, we can also determine the applicable elastic modulus when an external force is applied. This implies that by measuring these fluctuations, we can determine the temperature of the system that is causing the fluctuations if the elastic modulus is known; conversely, if the temperature is known, the elastic modulus can be determined. This will be discussed in the next section using specific examples.

Coffee break: A talk about making pies—an example of chaos

Knead the flour to make a candy pie, of 10 cm width and 2 cm height. Placing a red cherry somewhere on the pie (denoted as position x₁), tightly stretch the pie sideways to a length of 20 cm, and fold it in the middle. The height of the pie will again be 2 cm. If we repeat this process of stretching and folding the pie multiple times, the position x of the cherry keeps changing from the original position x₁ to a later position x₂ after folding. “What type of function of x₁ can be used to represent x₂?” is the most typical example of chaos^6.22, which is used even today.

I recommend you to recreate this experiment using a graph paper. The position of the cherry changes when you repeatedly stretch and fold the paper. The successive positions of the cherry should then be recorded. Because it is difficult to record too many details, you can just note whether the position is in the left or right half of the paper. Taking the right half as A and left half as B, we obtain a string of characters, such as A, B, A, B, etc. After acquiring 20 such records, we may obtain a sequence, such as A, A, A, B, B, A, B, A, B, A, B, B, B, B, A, A, A, B, B, A. Examining this sequence, we can identify the first position of the cherry. In other words, if the progress up to now is known, the initial condition can be specified. However, we have no idea whether the cherry will move to the right or left when we stretch and fold the paper for the 21st time from the 20 sequence. This example has been previously used to illustrate chaos, even though the word itself may not have been used at that time^6.23. In a text written by my classmate in the Department of Physics, he states that “the law of cause and effect is that, for the current situation, there is always a cause that can be traced back, but it does not tell you about the future”^6.24. I too thought so. In the pie kneading experiment, we cannot determine whether the cherry would move to the right or left after folding. Because everyone knows that nobody knows what will happen in the future or even tomorrow, I prefer to introduce this topic for physics lectures geared toward non-physics students in art college.

Acknowledgements

We thank Damien Hall (Kanazawa University) for providing an additional scrutiny of the contents of our translated manuscript.

Notes

6.1 Translator’s note: “Application of Statistical Mechanics to Phenomena in Cellular Systems” in Table of Contents.

6.2 Translator’s note: This lecture was delivered in 1996.

6.3 Translator’s note: “Low viscosity motion (in gas)” in Table of Contents.

6.4 Translator’s note: Please see Section 2-6: “It is because of the others that I have fewer gaming chips”. Please also see the figure in Translator’s note 2.9.

6.5 Translator’s note: Here the author mentions “viscosity”, although “density” is more commonly used for gases.

6.6 Translator’s note: “High viscosity motion (in liquid)” in Table of Contents.

6.7 Translator’s note: This is “kinetic energy” in the original Japanese version.

6.8 Translator’s note: This is “elastic energy” in the original Japanese version.

6.9 Translator’s note: This is only an analogy and the actual details are different.

6.10 Original note: In a 1940 paper [6.17] by Hendrik A. Kramers (1894–1952), a chemical reaction was qualified (without assuming thermal equilibrium a priori) as a process, in which a virtual particle representing the state of the system crosses the barrier between two wells via Brownian motion along the X-direction.

6.11 Translator’s note: The author is introduced in the original Japanese book.

6.12 Original note: Many chemical reactions in life processes utilize the energy released when adenosine triphosphate (ATP) decomposes into adenosine diphosphate (ADP) and a phosphate group.

6.13 Original note: The isolation of myosin is very slow, averaging approximately 20 s, whereas its interaction with F-actin is 100× faster.

6.14 Original note: Lord Rayleigh, John William Strutt (1842–1919). A British physicist. He accomplished extensive achievements in Rayleigh scattering of light, blackbody radiation, the discovery of argon; he researched Rayleigh number and capillary phenomena in fluid mechanics. He received the 1904 Nobel Prize in Physics.

6.15 Translator’s note: Eq. 6.5 was derived assuming the Maxwell’s distribution of molecular velocities.

6.16 Original note: Kodi Husimi (1909–2008). A Japanese theoretical physicist. After significant achievements in the field of statistical mechanics as a professor at Osaka University and Nagoya University, he became the first director of the Plasma Research Institute and a member of the House of Councilors, Japan; later on he made efforts to improve the research environment in Japan and overseas.

6.17 Original note: John G. Kirkwood (1907–1959). An American theoretical physicist. He accomplished great achievements in a wide range of fields, such as statistical physics, fluid mechanics, thermodynamics, and physical chemistry.

6.18 Original note: This series lasted until VIII in 1954.

6.19 Original note: Eugen Kappler (1905–1977). A German physicist. After studying the Brownian motion of a micro-rotating pendulum, he studied precious metals, while rebuilding the graduate school destroyed by World War II as a professor of experimental physics at the University of Münster.

6.20 Original note: There is a brief explanation about Kappler’s experiment in the Special Issue on Noise (1967) published by the Physical Society of Japan [6.16]. There is also an explanation in Reference [6.6].

6.21 Original note: Albert Einstein (1879–1955). A German theoretical physicist. He laid the foundation for the theory of relativity. He also won the 1921 Nobel Prize in Physics for his theoretical elucidation of the photoelectric effect.

6.22 Original note: A phenomenon that exhibits a complex state, in which the behavior is unpredictable even though the initial state and subsequent laws of motion are known. The “butterfly effect”, which states that even a small difference initially makes a big difference, is well-known.

6.23 Original note: A book “Theory of Probability and Statistics” [6.9] uses the term “group disorder”. The pie kneading experiment in this book was mainly used as an example to intuitively derive ergodicity (see Appendix B).

6.24 Original note: For example, pages 163 to 164 of Reference [6.7]: “The principle of causality states that if there is a certain proposition or an event, there will be a cause for it. However, the opposite is not the principle of causality. Causal principle does not state that there will always be the result if there is a cause. Please pay attention to this point. Going back to the cause from the result is different from telling the result from the cause.”

References

• [6.1]   Nagashima,  H.,  Asakura,  S. Dark-field light microscopic study of the flexibility of F-actin complexes. J. Mol. Biol. 136, 169–182 (1980). https://doi.org/10.1016/0022-2836(80)90311-3
• [6.2]   Kramers,  H. A. Brownian motion in a field of force and the diffusion model of chemical reactions. Physica 7, 284–304 (1940). https://doi.org/10.1016/S0031-8914(40)90098-2
• [6.3]   Fleming,  G. R.,  Courtney,  S. H.,  Balk,  M. W. Activated barrier crossing: Comparison of experiment and theory. J. Stat. Phys. 42, 83–104 (1986).
• [6.4]   Schroeder,  J.,  Troe,  J. Elementary reactions in the gas-liquid transition range. Annu. Rev. Phys. Chem. 38, 163–190 (1987). https://doi.org/10.1146/annurev.pc.38.100187.001115
• [6.5]   Sumi,  H. New developments in chemical reaction theory: Slow thermal fluctuation effect of solvated states. Butsuri 46, 911–918 (1991). https://doi.org/10.11316/butsuri1946.46.911
• [6.6]   Funatsu,  T.,  Harada,  Y.,  Tokunaga,  M.,  Saito,  K.,  Yanagida,  T. Imaging of single fluorescent molecules and individual ATP turnovers by single myosin molecules in aqueous solution. Nature 374, 555–559 (1995). https://doi.org/10.1038/374555a0
• [6.7]  Husimi, K. Theory of probability and statistics. Modern engineering publishing (Gendai Kogaku Sha, Tokyo, 1998).
• [6.8]   Kirkwood,  J. G. The statistical mechanical theory of transport processes I. General theory. J. Chem. Phys. 14, 180–201 (1946). https://doi.org/10.1063/1.1724117
• [6.9]   Kirkwood,  J. G. The statistical mechanical theory of transport processes II. Transport in gases. J. Chem. Phys. 15, 72–76 (1947).
• [6.10]   Kirkwood,  J. G.,  Buff,  F. P.,  Green,  M. S. The statistical mechanical theory of transport processes. III. The coefficients of shear and bulk viscosity of liquids. J. Chem. Phys. 17, 988–994 (1949).
• [6.11]  Kubo, R. Statistical mechanics (new format, In Japanese). Kyoritsu Shuppan Co., Ltd., Tokyo, Japan (2003).
• [6.12]   Kappler,  E. Versuche zur Messung der Avogadro-Loschmidtschen Zahl aus der Brownschen Bewegung einer Drehwaage. Annalen der Physik 403, 233–256 (1931). https://doi.org/10.1002/andp.19314030208
• [6.13]   Einstein,  A. Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Ann. Phys. 322, 549–560 (1905). https://doi.org/10.1002/andp.19053220806
• [6.14]   Ono,  S. Physics of noise. Buturi 22, 819–823 (1991). https://doi.org/10.11316/butsuri1946.22.819
• [6.15]  Yonezawa, T. Brownian motion. (Kyoritsu Shuppan Co., Ltd., Tokyo, 1986).
• [6.16]  Yanase, M. Philosophy of science. (Iwanami Shoten, Tokyo, 1986).
• [6.17]   Kramers,  H. A. Brownian motion in a field of force and the diffusion model of chemical reactions. Physica 7, 284–304 (1940). https://doi.org/10.1016/S0031-8914(40)90098-2