Chapter. Thermal energy: a minnow, an E. Coli an ubiquinone a) Consier a minnow using its fins to swim aroun in water. The minnow must o work against the viscosity of the water in orer to make progress. The water s viscosity creates a rag force on the minnow that issipates its energy. For a sphere moving through water the rag force was erive from hyroynamics by George Stokes in 1851 [1]. The formula for this force is F 6 v where is the viscosity of the water, typically one centipoise (0.01 gm/cms), is the raius of the sphere an v is the sphere s secular velocity. The minus sign reflects the fact that this force opposes the motion an ecreases the velocity as is seen with Newton s secon law that reas: M v F 6 v t in which M is the mass of the sphere. This simple equation has an exponentially ecaying solution with a relaxation time,, given by M 6 For the minnow these formulas must be moifie because the minnow is not a sphere. The minnow will instea be moele as a prolate ellipsoi of revolution with semi-major axis an semi-minor axis r. The rag force in this case is F 6 rkv
where K is given by [] K 1 ln 1 1 4 1 3 in which is given by r which is greater than one for a prolate ellipsoi. In the appenix, two limits of this formula are iscusse. When 1 it can be shown that K 1, an when 1, it can be shown that 4 6 rk ln r The first formula agrees with the rag force for a sphere, as it shoul, an the secon agrees with an expression that can be foun in a book by Howar Berg [3], an is applicable to something cigar shape, like a minnow. The rag force given above for a prolate ellipsoi of revolution is for motion parallel to the major axis as woul be the case for the swimming minnow. The minnow s fins create a swimming force, F, that works S against the rag force. Newton s secon law for the swimming minnow is 1 M t v 6 rkv F S If the swimming is steay, the swimming force will be constant an the minnow will achieve a steay state constant velocity given by
v F S 6 rk In this book, this steay state velocity is what is referre to as secular motion. The importance of viscosity in a hyroynamic context is etermine by a imensionless parameter calle the eynols number, [4]. It is efine in the present context by e e v where is the viscous iffusivity an is efine by in which is the mass ensity of the water. Clearly, the units for the viscous iffusivity are cm /s, the same as for the numerator of the eynols number. Water has a ensity close to 1.0 gm/cm 3. Thus the viscous iffusivity is approximately 0.01 cm /s. Assume that is about 8 cm an that the steay state velocity is about 100 cm/s. This is about ¼ miles per hour. Together, these quantities yiel a eynols number of 80,000. This is a very high eynols number (common usage is to speak of high an low eynols numbers rather than of large an small ones). Another important quantity to consier is the thermal spee of the center of mass of the minnow. As will be shown, this is a negligible spee for the minnow, but for the E. Coli an the ubiquinone it will be much larger. The thermal spee (in one imension), v, is efine by T v T kt M
The fact that the water in which the minnow swims is warm means that there are thermal fluctuations that perturb the motion of the minnow s center of mass. On the average this motion is characterize by the thermal spee. The volume, V, of the minnow, treate as a prolate ellipsoi of revolution, is V 4 r 3 Above, was taken to be 8 cm an r will be taken to be cm. This makes the volume equal to 134 cm 3. The minnow is at neutral ensity relative to the water which implies that its mass, M, is 134 gm. This makes the thermal spee for a temperature of 5 o C (we choose this temperature, equivalent to 98.15 K, throughout as characteristic unless specifie otherwise) equal to 1.75 x 10-8 cm/s. This is not quite Angstroms per secon an shoul be compare with the secular velocity of 100 cm/s. The thermal spee is ten orers of magnitue slower than the secular velocity. The secular velocity is sustaine by swimming an can be maintaine in a straight line for at least several secons. The thermal spee, on the other han, is prouce by myrias of collisions between the minnow an the water molecules that surroun it. An iniviual water molecule experiences collisions on a time scale of orer 10-14 s. Thus the minnow, in contact with huge numbers of water molecules, experiences collisions even more frequently. Since these collisions are from all irections an are uncorrelate the thermal spee oes not persist in a given irection for even a time as long as 10-14 s. This phenomenon gives rise to the concept of a mean free time, the time between collisions. The mean free path is the prouct of the mean free time an the thermal spee an in this case is increibly short, less than 10 - cm. The thermal spee shoul be thought of as a magnitue, i.e. a spee, an not as a velocity, i.e. a vector with irection since the irection oes not persist for times of any consequence as far as the minnow is concerne. The extreme values reporte here are the reason why the thermal motion of the minnow s center of mass is almost never mentione an is of no consequence in any case. It is mentione here for purposes of comparison with the other cases to be escribe later in this chapter.
The consequences of a short mean free path an a small mean free time can be capture by the concept of iffusion. For times long compare to the Langevin relaxation time, the Brownian motion can be escribe by iffusion [5]. The iffusion equation in one spatial imension associate with the Langevin equation is given by t f ( x, D x f ( x, where f ( x, x enotes the probability for fining the Brownian particle between x an x x at time t, an D is the iffusion constant with the units cm /s an is given by Einstein s formula kt D 6 rk This result is erive in appenix.. It may be shown that the mean square eviation of a Brownian particle in time t is given by x Dt For the minnow the value of the iffusion constant is Thus the root-mean-square isplacement in one secon, given by 14 D 6.8 10 cm / s. Dt, is 3.7 x 10-7 cm (3.7 nm). In one secon the secular isplacement is 100 cm. The iffusion of the minnow s center of mass is entirely negligible compare to its secular motion. Because the thermal isplacement epens on the square-root of time an the secular isplacement epens on time linearly, there is a time for which both are equal. This time works out to be 1.36 x 10-17 s. For all longer times the secular isplacement is larger an for one secon it is 15 orers of magnitue larger, as was just shown above. A eynols number can be associate with the thermal motion of the minnow as well as with the secular motion. Instea of using the steay state
secular velocity in the numerator of the eynols number ratio, use the thermal velocity. This gives a eynols number of e v T 1.410 6 which is a very low eynols number. In escribing the secular motion of the swimming minnow the Stokes formula for the rag force on a prolate ellipsoi was use. This formula is vali only for sufficiently slow motions. The first orer Oseen correction to the Stokes formula for a sphere is [6] F 3 6 v 1 8 e which implies that the simple Stokes formula, without the 3/8 e correction, only works for low eynols number. For the secular motion of the minnow for which the eynols number is 80,000, the true rag force is much more complicate, is nonlinear in the velocity an involves a turbulent wake behin the minnow. However, for the thermal motion the Oseen correction is completely ignorable since the thermal eynols number is so low. For all of the other cases treate in this book, both for secular an for thermal motion, the eynols number is low an the Oseen correction is not neee. b) An E. Coli is a small cylinrical bacterium about microns (m) ( x 10-4 cm) long an with a cross-sectional iameter of about 1 m. Its mass is 1 about M 10 gm. In the environment in which it typically swims the viscosity is 0. 07 poise (0.07 gm/cm-s). The E. Coli is able to swim because it usually has six flagellar filaments that emanate from ranom positions on its boy an exten into the surrouning meium about three boy lengths [7]. Viewing the E. Coli from behin as it swims away the flagellar filaments rotate counter-clockwise an form a synchronous bunle that propels the cell boy forwar. This bunle can rotate up to 40 times per secon. Since the rotation is in a viscous meium there is a reaction torque on the E. Coli boy that causes a clockwise rotation of the boy, but at a
much slower rate than the rate for the flagellar bunle. This is because the E. Coli moment of inertia is much larger than that of the flagellar bunle. The E. Coli can continue its secular run for an average time of 1 secon. It then stops, reverses the flagellar filament rotation irection an reorients its irection. When the filaments are rotate in reverse, i.e. clockwise, they o not form a single bunle but instea stick out in ifferent irections an the effect of their rotation is to reorient the irection of the E. Coli in an essentially ranom way. This motion is calle a tumble. The tumbles take, on average, about 0.1 s. The runs are the secular motion of the E. Coli an 3 provie it with an average spee of v 10 cm/s. Therefore, the eynols number for the secular motion is e 3 v 10 10 0.07 4 7.4 10 6 which is a very low eynols number. In this calculation the semi-major axis is 10 4 cm an the flui meium mass ensity is 1.0 gm/cm 3. Note that this secular eynols number is 10 orers of magnitue smaller than that for the minnow. This means that the motion of the E. Coli is ominate by viscosity whereas that of the fish is ominate by inertia. Using the Stokes rag force with the prolate ellipsoi of revolution correction, Newton s secon law for the E. Coli is given by M v 6 rkv t F S where 4 r 0.5 10 cm is the semi-minor axis an for the value of the correction factor is K 1. 04(the Berg approximation given in the appenix yiels 1.504 for this case). This means that the relaxation time for the E. Coli motion is M 6.5310 6 rk 8 s
Therefore, if the E. Coli stops swimming while moving with a velocity of 3 v 10 cm/ s, then it travels a istance of 0 t v 0 0 t exp v 0 1.310 10 cm before it stops [7]. This is much less than an Angstrom!! This result emphasizes the meaning of viscosity ominating motion for low eynols number. However, it is not the whole story because of the Brownian motion that as a ranom motion to this secular behavior. The thermal spee for the E. Coli is given by kt vt 3 0.4 cm / s M This is a hunre times bigger than the secular velocity!! However, the secular velocity is maintaine in virtually a straight line for up to a secon whereas the thermal spee is maintaine in a given irection for much less than 10-14 s. Another way to see this to look at the root-mean square eviation, as was one for the minnow. This requires the iffusion constant, D, given by kt D 1.36 10 6 rk Thus, the root-mean-square eviation, give by 9 cm s Dt, equals 5 x 10-5 cm for a time of one secon. The secular isplacement in a secon is x 10-3 cm given the average spee for a run an the fact that runs last on average a secon. The secular isplacement is 40 times the thermal isplacement. While this is a large ratio it is not enormous like it was for the minnow. The time for which these two isplacements are equal is 0.68 x 10-3 s. For times longer than this the secular isplacement ominates. However, this result shows that thermal motions are significant at the ms time scale, the time
scale for many chemical processes in the E. Coli s metabolism. Note also that even though the thermal spee is larger than the secular velocity, the thermal eynols number is still much less than one. c) Ubiquinone is a ubiquitous component of electron transport chains in aerobic bacteria, mitochonria an chloroplasts. It is also calle coenzyme Q, or CoQ for short. Below the notation UQ will be use for the generic quinone. Its ubiquity in aerobic organisms gives it the name ubiquinone. In the most common form in mammalian mitochonria the quinone ring is attache to a tail containing ten isoprene units. This tail facilitates the solubility of UQ in the mitochonrial membrane s lipi interior where UQ carries out its function. In chloroplasts it is calle plastoquinone an the isoprene tail may contain 6-10 units. UQ s role is as an intermeiary of reox reactions in the electron transport chain. In this role in unergoes a reox cycle in which the two carbonyl oxygens of the quinone ring become successively reuce (to hyroxyls) an oxiize. These oxiationreuctions involve whole hyrogen atoms, i.e. an electron an a proton for each oxygen. Thus the molecular weight of the oxiize form is 86 an that of the reuce form is 864. This makes the mass of UQ or UQH 1.44 x 10-1 gm. This is nine orers of magnitue smaller than the mass of the E. Coli, an 3 orers of magnitue smaller than the mass of the minnow. The electron transport chains are the means by which carbohyrate substrates are oxiize by oxygen in metabolism. Some energy harvesting takes place uring glycolysis an uring the citric aci cycle, but the majority of the harveste energy is extracte by the electron transport chain through a combination of chemiosmosis an membrane assiste ATP synthesis. This is manage by a series of oxiation-reuction reactions involving iron-sulfur proteins, cytochromes, a few other coenzymes an UQ/UQH. While the etails of these processes are fascinating, they take us beyon the scope of this chapter. What is important here is the fact that all of these components, except for UQ/UQH, are embee in the membrane by a process of self-assembly an maintain their relative positions to each other an with respect to the inner an outer surfaces of the membrane. The terminology, inner an outer membrane surfaces refers to the lipi bilayer
that makes up the membrane. Since the intact membrane is a close surface, it separates an interior region from an exterior region. Thus, the membrane s inner surface is ajacent to the interior of the membrane surroune compartment, an the membrane s outer surface is ajacent to the external environment surroune the membrane compartment. In aition, the lipi interior of the membrane bilayer itself will be referre to as well. This is where the enzyme complexes are embee as well as where UQ/UQH moves about. Only the UQ/UQH are freely mobile species that iffuse back an forth between electron onors at the membrane interior interface an electron acceptors at the membrane exterior interface. Two types of complexes are present. The first is largely mae up of iron-sulfur proteins an process reuce NADH that is the electron onor supplie primarily by the citric aci cycle. This complex is calle NADH-Q reuctase. The secon is largely mae up of cytochromes an has molecular oxygen as its ultimate electron acceptor. It is really two complexes, calle cytochrome reuctase an cytochrome oxiase. Many replicas of these enzyme complexes exist in a given membrane an many UQ/UQH s are also present to connect the two segments of the transport chains. The focus here is the mechanism by which UQ/UQH perform their function. This mechanism will serve as a paraigm for rectifie Brownian motion as a general mechanism in subcellular biology. [See pp. 30-3 of: Lipis, membranes an chemiosmosis] The carbonyl oxygens of the quinone ring are polar groups, both when oxiize an when reuce to hyroxyl groups. As such they woul not be very soluble in the interior of the membrane lipi bilayer. The isoprene tail, however, is lipophilic an flexible. It is reasonable to assume that the tail wraps aroun the quinone ring an shiels the polar groups from the lipi environment in which UQ/UQH move. This means that UQ/UQH, whether oxiize or reuce, may be moele as a sphere of raius 7.5 Angstroms, or 0.75 nm given its mass. Membranes of the lipi bilayer structure are between 60 an 100 Angstroms thick. Since the UQ/UQH has a raius its center s motion across the membrane will be assume to cover about 80 Angstroms, if a membrane thickness of 95 Angstroms is assume an room
is left for the size of the UQ/UQH. The reox potential change from the electron onor sie to the electron acceptor sie for UQ is of the orer of 0.1V (Volts) uring steay state metabolism. Since UQ is reuce by two electrons (an two protons) the change in Gibbs free energy is given by -0.1 x x 1.6 x 10-19 J per molecule. Converting to Kcal requires multiplication by 10-3 /4.18, yieling a final result of 7.7 x 10-4 Kcal. To get this in Kcal per mol requires a factor of 6 x 10 3, yieling 4.6 Kcal/mol. This is not very much energy an as will be seen it has nothing irectly to o with the energetics of actually getting the UQ across the membrane. Picture a cut through the membrane oriente so that the interior compartment is on the left an the external environment is on the right. UQ is reuce near the left sie of the membrane by electron onors an is calle UQH. Near the right sie of the membrane, electron acceptors oxiize UQH back into UQ, the oxiize form. These quinone species move in between through the lipi bilayer interior of the membrane. This interior has a viscosity of about 5 cp (0.5 gm/cm-s). Their ynamic motion is governe by the Langevin equation with Stokes rag force given for a sphere. M t v 6 v F ~ ( wherein the mass, raius an viscosity are given above an the fluctuating force satisfies the fluctuation-issipation relation given in appenix.. For this case the Langevin relaxation time is given by M 6 4.07 10 15 s This is a very short relaxation time an if it is much shorter than the time for UQ/UQH to cross the membrane then the Langevin escription of the motion can be replace by iffusion in accor with appenix.. This conition can be etermine self-consistently by assuming that iffusion is a satisfactory escription an computing the root-mean-square isplacement
formula to get the average time of transit across the membrane. The Einstein formula for the iffusion constant is kt D 1.16 10 6 Assuming that UQ/UQH travel a istance for membrane crossing is given by 7.7510 D 6 t D cm s 8 nm, the iffusion time, t D, Since this is 9 orers of magnitues longer than the Langevin relaxation time, the escription is in the extreme iffusion limit of the Langevin equation. While this may seem surprising for such a small istance as 8 nm, it is a result of looking at a molecule rather than at some macroscopic object. The Langevin relaxation time is so short because the mass is so small. A better approach to this issue is to calculate the so-calle mean first passage time (MFPT) for the quinone to go across the membrane. In this instance the result is precisely the same as given above. The stage is now set to consier rectifie Brownian motion for the quinone shuttle. Ubiquinone (UQ) receives a pair of electrons from FeSproteins situate near the insie surface of the bacterial membrane at the same time that it receives a pair of protons from the cytosol of the bacterium. These two pairs reuce UQ to UQH. This reuce species of quinone is free to iffuse insie the lipi bilayer interior of the bacterial membrane. Eventually, in a few microsecons as was compute above, it arrives near the outsie surface of the bacterial membrane where it gives up its electrons to cytochromes an its protons to the external milieu in which the bacterium lives. Having one so, the quinone is re-oxiize an UQH becomes UQ again. This species is free to iffuse insie the lipi interior of the bacterial membrane an eventually makes its way back to near the insie surface for another roun of reuction an oxiation. As long as there are electron onors near the insie surface an electron acceptors near the outsie surface s
there will appear, on the average, to be a steay cycle of UQH moving from insie to outsie an UQ moving from outsie to insie while shuttling both electrons an protons. The electrons travel own the electron transport chain an the protons traverse the membrane from insie to outsie. Thus, the electron onors an acceptors create asymmetric bounary conitions for the quinone iffusion by changing the ientity of ubiquinone from reuce to oxiize to reuce etcetera. This is the essence of rectifie Brownian motion. This escription can be mae quantitative by explicitly using the iffusion equation. Let f ( x, enote the probability ensity at time t for reuce UQH an let g ( x, enote the probability ensity at time t for oxiize UQ. The insie surface of the membrane is locate at x 0 an the outsie surface is locate at x (in the figure x = L is use). In steay state it is expecte that the probability ensity for UQH at the insie surface, enote by Q, an the probability ensity for UQH at the outsie r in r r r surface, enote by Q, satisfy Q > Q, because UQH out in out is prouce at the insie surface an is converte at the outsie surface. Similarly, in steay
state it is expecte that the probability ensity for UQ at the insie surface, enote by Q, an the probability ensity for UQ at the outsie surface, enote by Q o in o out, satisfy o Q > out o Q, because UQ is prouce at the outsie in surface an is converte at the insie surface. The reuce species satisfies the iffusion equation t f ( x, D x f ( x, with the bounary conitions at steay state given by f (0) Q an f ( ) Q r in r out where the subscript enotes the steay state values. Similarly, the oxiize species satisfies the iffusion equation g( x, D t x g( x, with the bounary conitions at steay state given by g (0) Q an g ( ) Q o in o out These equations are easily solve an have the steay state solutions f g ( x) f ( x) g x (0) x (0) f g (0) f (0) g The probability currents, or fluxes, are efine by ( ) ( )
D D f ( x) f (0) f ( ) 0 x D D g ( x) g (0) g ( ) 0 x wherein the left-han sies efine the fluxes in the manner that is stanar for iffusion (see appenix 1.1), an the right-han sies are the results for the particular steay state solutions given above. The inequalities result from the bounary conitions. The meaning of these fluxes is simple, the reuce species goes from 0 to an the oxiize species goes from to 0. The flux magnitue is etermine by D 1.16 10 7 810 7 cm s 0.145 Note that the same D is use for both UQ an UQH because they are of only slightly ifferent raii, iffering in mass by only altons out of 86. As long as energy metabolism is functioning so that the electron onors an acceptors maintain the asymmetric bounary conitions for reuce an oxiize quinone species, these non-zero fluxes are unchange. If metabolism is shut own, then the bounary conitions become symmetric an the fluxes vanish. Consier the thermal spee of the quinones. It is given by cm s v T kt M 5.34 10 3 cm s This is equivalent to 10 miles per hour. If the quinones coul move in a straight line at this spee, they woul cross the 8 nm membrane in 1.5 x 10-10 s. This is four orers of magnitue faster than the iffusion time calculate above. The ifference is accounte for by the fact that the Brownian motion is not sustaine in a straight line for even as long as 10-14 s since the mean free time is less than this. Brownian motion is an erratic back an forth motion over very short steps. Thus, the path from one sie of the membrane to the other by Brownian motion is not a straight path but is instea mae up
of very many short back an forth steps so that the total Brownian path length actually traverse by the quinones is, on the average, of orer 80 m. Unlike the minnow or the E. Coli, the quinones have no other means for motion than Brownian motion. There are no fins nor flagella for the quinones. Since the reox states of UQ an UQH are both electrically neutral, there is no electrical force cause by the electrical potential ifference across the membrane create by chemiosmosis in actively metabolizing cells an organelles. This means that at the subcellular molecular level, rectifie Brownian motion is perhaps the only source of effectively secular motion. Appenix.1: Drag force correction formula For 1 the prolate ellipsoi of revolution is simply a sphere after all an the correction factor, K, shoul equal 1. To see that this is so, replace by 1 where 0 for the prolate case. The esire limit is the limit 0. In this limit, replace by 1 everywhere in K. The key step in taking the limit is in properly expaning the logarithm term to thir orer: 1 1 3 ln 1 1 3 1 1 3 When this is combine with the other factors in the enominator, both the 8 enominator an the numerator become as 0. 3 The other limit is the highly prolate limit for which 1. In this 1 case it is easier to work with, in which case 1. K can now be written in the form 3
For small it follows that K 1 4 1 3 1 1 ln 1 ln 1 1 ln 4 1 ln ln 4 Combining this with the other factors an ropping all terms yiels 1 K 3 1 ln This verifies the rag formula quote from Berg in the text above. For a value of 0. 5 which will be use for the minnow, the exact formula for K gives 1.598 whereas the approximation gives 1.688. This is not too ba since is not really much less than 1.0. Thus the exact formula will be use. In the E. Coli case, 0. 5 is use an the approximation is quite ba. Appenix.: erivation of the iffusion equation Start from the generic Langevin equation for a sphere of raius M t v 6 v F ~ ( ~ in which F ( t ) is a fluctuating force that represents the statistical properties of the myrias of collisions between the water molecules an the Brownian particle. An exact ynamical solution to the problem of the motion of the Brownian particle an all of the flui molecules by Newton s laws is simply intractable. By introucing a fluctuating, or stochastic force, Langevin was
able to capture the essential statistical properties in a phenomenological ~ equation. The statistical properties for F ( t ) are that it is a Gaussian process with first an secon moments given by ~ F( 0 ~ ~ F ( F ( kt6 i j ij ( t The first moment equation implies that on the average the fluctuating force has no effect since collisions are equally likely from all irections. The secon moment equation is more complicate in meaning. The force subscripts enote the Cartesian components an the Knonecker elta of these subscripts implies that the ifferent Cartesian component fluctuate inepenently of each other. The Dirac elta function of time means that the correlations are very short live. The correlation coefficient contains two types of terms. The first term is the thermal factor, kt, that implies that the amplitue of the fluctuations increases with temperature. The secon factor is ientical to the Stokes rag coefficient an is the basis for what is calle the fluctuation issipation relation that intimately connects the fluctuation strength with the secular relaxation parameter. When the relaxation time is very short, or when the ynamics is at very low eynols number, then the inertial term in the Langevin equation can be neglecte an the equation becomes an since this reuce equation is equivalent to ~ 0 6 v F( x v t ~ x F( t 6
This stochastic ifferential equation can be written as x ~ g ( t ) t in which ~ ~ g ( t ) inherits statistical properties from F ( given by ~ g ( 0 ~ ) ~ kt g ( t g ( t i j ) 6 ij ( t To every such stochastic ifferential equation there is associate a Fokker- Planck equation [8] that in this case reas kt P( x, t 6 P( x, This is clearly a iffusion equation with a iffusion constant given by Einstein s formula. When it is restricte to one imension the Laplace operator is replace by a secon orer erivative in x. eferences [1] Lamb, H., Hyroynamics, (Dover, New York, 1945), chapter XI. [] From an internet article by G. Ahmai. [3] Berg, H., anom walks in biology, expane eition, (Princeton U. Press, Princeton, N. J., 1993), p.57. [4] Berg, H., op. cit., p. 75. [5] Einstein, A., Investigations on the Theory of the Brownian Movement, (Dover Publications, New York, 1956)
[6] Lanau, L. D. an E. M. Lifshitz, Flui Mechanics, (Pergamon Press, Lonon, 1959), p. 68. [7] Berg, H., op. cit., chapter 6. [8] van Kampen, N. G., Stochastic Processes in Physics an Chemistry (North-Hollan, Amsteram, 1981).