`
`256 [17] ‘In Vitro PROTEIN DEPOSITION
`
`
`
`[17] Analysis of Protein Aggregation Kinetics
`
`By FRANK FERRONE
`
`This article takes up the question of describing the formation of large
`aggregates of proteins ordered by specific contacts. There are two goals in
`any modeling of protein aggregation. Thefirst is to validate a possible
`mechanism by reducing a given proposal to a kinetic scheme whosepredic-
`tions (time course, concentration dependence,etc.) can be compared with
`experiments. The second goal is to determine the molecular ingredients
`once a viable scheme has been established: what the rate constants are and
`what determines their observed values.
`In following a kinetic process, modeling the initiation of the process
`poses the greatest challenges. This is because the latter stages are usually
`more amenable to direct observation, whereasthe initial phases are more
`likely to be controlled by intermediatesthat aredifficult to observe directly.
`Although protein aggregation has been studied for quite a long time, a
`numberoffallacies persist. Probably the most notable is the assumption
`that a lag time in the kinetics represents a nucleation phase and that the
`end of such a lag corresponds to cessation of nucleation. Thisarticle first
`develops a basic description of the association process using a setof fairly
`reasonable assumptions and then turns to more advanced topics in the
`sections that follow. While it is impossible to cover all the possible mecha-
`nisms or their ramifications, the goal of this article is to provide a robust
`methodof attacking all types of assembly process, keeping in mind the two
`goals of establishing mechanism and determining rates.
`This article concentrates on the description of net properties, such as
`total mass polymerized, for aggregations at times neartheir initiation,e.g.,
`during the first 10% or so of the reaction. There are a number of other
`fascinating topics such as polymer length redistribution, which will not be
`discussed here. The readeris referred to the discussion of other relevant
`work throughout the article. We will also omit any systems for which the
`polymerization is coupled to energy sources, such as the hydrolysis of ATP
`or GTP.
`
`Basic Description of Aggregation
`
`Protein aggregation processes are formally represented by the addi-
`tion reaction
`
`A+ A, = Ana
`
`(1)
`
`METHODSIN ENZYMOLOGY, VOL.309
`
`Copyright © 1999 by Academic Press
`All rights of reproduction in any form reserved.
`0076-6879/99 $30.00
`
`Amgen Exhibit 2043
`Apotex Inc. et al. v. Amgen Inc. et al., IPR2016-01542
`Page 1
`
`Amgen Exhibit 2043
`Apotex Inc. et al. v. Amgen Inc. et al., IPR2016-01542
`Page 1
`
`

`

`
`
`{17] 257 PROTEIN AGGREGATION KINETICS
`
`
`
`This reaction is, in principle, characterized by the knowledge ofall rate
`constants for elongation and depolymerization for species of all size, i.e.,
`the knowledgeofall values of k; and k;,. A plausible goal therefore might
`be to determinethe individual reaction step rates for all n values, and given
`the ready accessibility of computers that can perform numerical integration
`of rate equations, such a brute force model is easy to construct and solve
`numerically. If a comprehensive table of rate constants were to be con-
`structed, one would immediately need to winnow the set down to an under-
`standable pattern, or, equivalently, in building a model, one would need a
`rationale for a particular pattern of constants.In short, somerule is required
`to make sense of the variousrates, if they were known, or to simplify the
`model if one is being constructed. This is especially true for the testing
`phase, where one is faced with the problem of varying 2n parameters to
`obtain their optimum values for an n step model.
`One immediate simplification occurs for long polymers.It is generally
`found that once n gets to be large, elongation and shortening becomesize
`independent, and one can describe the net elongation rate of polymers,
`denoted by J, in terms of the elementary rate constants as
`
`J=kon~k.
`
`(2)
`
`in which c is the concentration of monomers [A]. From Eq.(2) it is clear
`that a basic experimentconsists in the simple measurementof elongation
`or shrinkage of polymers, so as to measure J. If c is known,or ideally if
`the measurement is performed as a function of c, the elementary rate
`constants can be inferred.
`Most interesting assembly reactions do not begin with the samerate
`constants as they conclude. In a great many cases of interest, the initial
`reaction steps are slowerthan the later ones. In evolved systemsthis allows
`for control of the reaction because the initiation can be spatially localized,
`despite the general presence of favorable growth conditions. In some patho-
`logic systemsthis initial inhibition has the advantage of allowing the organ-
`ism to survive longer.! Whatever the reason, a fundamental question for
`any kinetic mechanism must be whetherit showssuch aninitial inhibition.
`It is a great help in understanding such a system if the initial reaction
`steps may be considered close to equilibrium, as the aggregates may be
`considered as thermodynamic species rather than kinetic intermediates.
`In terms of the preceding comments on inhibition, this equilibrium repre-
`sents a series of unfavorable equilibria, at least up until some point is
`reached,ie.,
`
`1A. Mozzarelli, J. Hofrichter, and W. A. Eaton, Science 237, 500 (1987).
`
`Page 2
`
`Page 2
`
`

`

`
`
`258 [17] In Vitro PROTEIN DEPOSITION
`
`
`
`A+A,——Apyat
`A+A;,— Ani
`
`n<n*
` n>n*
`
`(3)
`
`The equilibrium probability of finding a given species A, can be related
`to a Gibbs free energy (relative to some standardstate) as
`
`[An] = [Astandara] exp[—AG(n)/RT]
`
`(4)
`
`in which R is the gas constant, T the temperature in Kelvins, and [Astanaaral
`is the standard state concentration. One logical choice for this “standard
`state”’ is the initial monomerconcentration. Then the energy of each aggre-
`gate is measuredrelative to the initial concentration, although differences
`between curvesastheinitial concentration is varied would then be masked.
`Another common choice is some arbitrary concentration, say, 1 mM. Then
`the free energies are measuredrelative to a fixed concentration. Suchfixed
`standard states can produce a paradoxical result on occasion, namely that
`the aggregate may be favored relative to the standard state. This simply
`means that the arbitrarily chosen standard is too high. In any case, the
`decrease in probability in finding A,, relative to [Agtandara] Corresponds to
`an increase in energy. When a sequence of steps involves the increase in
`energy, the steps in the reaction can be viewed as climbing an energetic
`barrier that must be overcomefor the aggregation process to proceed (Fig.
`1). At equilibrium,
`
`kic[A,] = Kiv[Anwl
`
`(5)
`
`from which it follows that [A,+:]/[An] = cki/kj41. However, it is also
`clear that
`
`ra=e{+ feecesere(nt ty—n | /xr|
`
`
`= exp (-{et/rr)
`(6)
`
`[Ans] _
`
`AG(n + 1) - AG(n)
`
`In other words, the slope of the free energy plot, dAG/dn,is related to
`the ratio of the rate constants into and outof a state for a given monomer
`concentration c. The changeoverin ratesis therefore related to the change
`in slope of the free energy barrier, and a barrier that is linear with size
`gives a constantrate ratio. When the turning pointis sufficiently sharp, the
`implication is that there is one state with a particularly small population
`that will represent the rate-limiting step for the reaction. This bottleneck
`is a thermodynamic nucleus, a necessary but very scarce species in the
`reaction path. This is quite distinct from a structural nucleus, in which a
`
`Page 3
`
`Page 3
`
`

`

`
`
`[17] 259 PROTEIN AGGREGATION KINETICS
`
`
`
`AG(kcal/mol)
`
`o3 ao3c 0
`
`5
`
`10
`
`15
`
`20
`
`aggregate size
`
`Fic. 1. Typical free energy barrier. Free energy of the aggregate AG (relative to the
`monomer) is shown on the vertical axis, whereas size of the aggregate is shown on the
`horizontal axis. The nucleus is the species whose size corresponds to the peak of the energy
`curve, and thus for which the populationis smallest. Polymerization requires that the aggregate
`size pass through this maximum, which equates the reaction to a barrier crossing. The slope
`of the curve at any size n is controlled by the concentration times the ratio of rate constants,
`ck3/k,;41. To assumethat the rate constants are independentof 7 on eitherside of the nucleus
`would imply that the slope is the same for different sizes n, in turn implying that the free
`energyis linear with n in that range. At large values of n, this assumption is reasonable.
`
`specific stable structure fosters further growth. A thermodynamic nucleus,
`by its nature, is the least stable and hence least prevalent species in the
`reaction. While stable structural nuclei may also be scarce, nothing in the
`mechanism intrinsically requires their scarcity, and their stability permits
`a variety of strategies for their capture and study that are simply infeasible
`for unstable nuclei.
`When the small concentration of nuclei effectively form a barrier to
`further growth, then the rate of the formation of polymers is set by the
`population of nuclei and the rate of elongation of the nuclei themselves,
`ie., the rate of crossing this effective barrier. (For a more detailed treatment,
`see the section on Generalized Nucleation.) If c* is the concentration of
`nuclei and J* is the rate of elongation of the nucleus, then polymers at
`concentration c, are formedat a rate
`dc,/dt = J*c*
`
`(7)
`
`Page 4
`
`Page 4
`
`

`

`(17]
`In Vitro PROTEIN DEPOSITION
`260
`
`Note that polymers may formally be counted by their ends. Once polymers
`are formed, they add mass byaccretion to their ends. If the assumption is
`made that polymer additionis all by the same rates J which do not depend
`on size n, then if we call A the total concentration of monomers that have
`gone into polymers,
`
`(8)
`dAldt = Icy
`If all molecules must be classed as either polymers or monomers, then
`the original concentration c, becomessplit into these, and we can write,
`A(t) = co — e(t)
`(9)
`The accuracy of this separation into polymers and monomers depends on
`the rarity of intermediate species, and this should be a good approximation.
`J and J* clearly depend on ¢ as well. The solution of this set of Eqs.
`(7)-(9) is not simple, although given values for the various parameters,it
`is straightforward to construct a numerically integrated solution (see Fig.
`2). It is possible to obtain an analytic solution for such equations when the
`
`0.8
`
`os
`
`0.4
`.
`
`0.2
`
`0
`
`0
`
`0.5
`
`1
`
`1.5
`
`time (s)
`
`~C
`
`c 2
`
`a wC
`
`c S
`
`5
`
`4&
`
`Fic. 2. Nucleation-controlled aggregation kinetics, The solid curve is an exact, numerically
`integrated solution to Eqs. (7)-(9), in which it is assumed that J/* = k,c. In the numerical
`solution, it is not assumed that the forward rates are much greater than the reverse rates. At
`long times, the exact solution goes to 1. The long dashed curve, labeled t”, showsthe result
`of simply treating the monomer concentration as a constant, as described in Eq. (18). The
`short dashed curve, labeled cos, shows the result of using the linearized equations, as given
`by Eq. (12). Note that the cosine solution also begins as ¢* butis closer to the exact solution.
`
`Page 5
`
`Page 5
`
`

`

`
`
`[17] 261 PROTEIN AGGREGATION KINETICS
`
`
`
`forward rates significantly exceed the reverse,” as well as to deduce the
`concentration dependence from scaling arguments independentof the ac-
`tual solution of the equations.** We would like to introduce a different
`approach because of its general utility. The approach we take is known as
`a perturbation approach, whosecentral idea is to expand various quantities
`about theirinitial values in such a way that the resulting equations become
`linear and soluble. In this approach,one formally expandsall the equations
`abouttheir initial values. For example,
`
`J(c) = J(cy) + (dJ/dc),(e — ce.) +... . = 4 — (di/dce),At+....
`
`(10)
`
`where J, is defined as J(c,) and (dJ/dc), means the derivative is evaluated
`at c = Cy. Only lowest order terms are retained. Smallness is formally
`defined relative to c,, the initial concentration.
`Then Eqs. (7) and (8) become
`
`dc,/dt = J,*co* — [d(J*c*)/dc] A
`AAIAt = Iyepo
`
`(11a)
`(11b)
`
`Equation (11b) contains no higher terms because c,, begins as a small term
`intrinsically, in contrast to c,, which is the initial value of c and which is
`not smallat all.
`The solution to the just-described set of equations has the form
`
`A = A[1 — cos(Ba)]
`
`(12)
`
`Before examining this further, the reader should note that the cosineis
`only employed nearthe initial time, so that Br never becomes large and
`the oscillatory behavior of the cosine function is not seen.
`In terms of parameters that appear in the original rate equations,
`
`and
`
`JSCA=
`4 rex
`ac I *6*)
`
`B? = J[d(J*c*)/dc]
`
`(13a)
`
`(13b)
`
`? F. Oosawaand S. Asakura,“Thermodynamicsof the Polymerization of Protein.” Academic
`Press, New York, 1975.
`3R. F. Goldstein and L. Stryer, Biophys. J. 50, 583 (1986).
`4H. Flyvbjerg, E. Jobs, and S. Leibler, Proc. Natl. Acad. Sci. U.S.A. 93, 5975 (1996).
`5M.F. Bishop and F. A. Ferrone, Biophys. J. 46, 631 (1984).
`
`Page 6
`
`Page 6
`
`

`

`
`
`262 [17] In Vitro PROTEIN DEPOSITION
`
`
`
`The product of these twois also interesting, giving
`
`B’A =J,J#c%
`
`(13c)
`
`In Eq. (12), B is clearly the effective rate constant and 1/B is the time
`constant for the reaction. So we ask how theinitial rate constant depends
`on the concentration. Consider two cases. First suppose the nucleus is an
`aggregate of somesize n, so that
`
`c* = K,xc™
`
`(14)
`
`where K,,+ is used to indicate the equilibrium constant for association of
`n* monomers. Let us also assumethat the reverse rate k* can be neglected
`so that
`
`B? = (n* + 1)k#IKyec™
`
`(15)
`
`the forward rates are all greater than the reverse rates
`in fact,
`If,
`(ie., k* > k* and k , > k_), then B ~ c+), or the characteristic time
`(1/B) ~ c"+2, This is a familiar result”; i.e., a plot of log B vs log c has
`a slope of (n* + 1)/2, or perhapsslightly less if the depolymerization rates
`cannot be ignored. However, the salient feature of an equilibrium nucleus
`in a simple linear elongation reaction will be rates with higher than unity
`dependence on concentration.
`The other extreme occurs when the concentration of nucleiis fixed, as
`when they are provided by using preformed seeds. Returning to Eq.(11a),
`the concentration of nuclei is fixed in this case and therefore is not given
`by a monomerequilibrium. Mathematically, this means c* does not depend
`on c and hence dc*/dc = 0, and then
`
`B? = Jc*dI*ldc = k#c*(k,c — k_)
`
`(16)
`
`Nowthe concentration dependence of B is sublinear, and at best B will go
`as the square root of c! For either of these extremes, however, the time
`course of the reaction is similar. Its leading term is parabolic (i.e, t”). The
`effect of this parabolic initiation is to give a weak delay at the start of
`the reaction.
`Note that the shape of the curve is the same regardless of whether
`nuclei are formed. Such a curve is sometimes taken as indicative of nucleus
`formation, but as we have seen just now, even with a supply of preformed
`nuclei and no new nuclei created, the time dependencewill be the same as
`the case whennuclei are in equilibrium with monomers. For both preformed
`nuclei and thermodynamic nuclei, all nuclei appear at the start of the
`reaction, and the upward curvature has nothing to do with their formation.
`As wewill see later, it is possible to have a much more abrupt curve than
`
`Page 7
`
`Page 7
`
`

`

`
`
`[17] 263 PROTEIN AGGREGATION KINETICS
`
`
`
`t” (equivalent to a more pronounced delay). It is also evident that nu-
`clei do not disappear after the lag period, even in the case of thermody-
`namic nucleation. To verify this comment, consider that the concentra-
`tion of polymers may be obtained using Eq. (12) in Eq. (11b) to give
`cp = (BA/J,) sin(Bt). For nucleation to abate after the lag requires the
`concentration of polymers to fall, but from the foregoing it is clear that c,
`only abates slightly during the weak “lag phase.”
`If one observes such a shape of the time course (¢? or cosBt), what
`strategies are appropriate? Clearly it is helpful to know J, so that some
`methodof following the growth of polymers is desirable. A very effective
`way to do this is by differential interference contrast (DIC) microscopy.®
`Next, the parameters A and B that describe the growth are determined by
`curve fitting the initial growth phase. The concentration dependence of
`B’A [using Eq. (13c)] will give n*, the nucleus size. K,+ and J* remain to
`be determined and are more problematic as they may apply to the properties
`of the least populousspecies, the nuclei. (Of course, if nuclei are preformed,
`then the issue is quite different). It may be possible to relate K,,+ to other
`known equilibria of the system. As described later, k* can be neglected.
`Evenso,oneis left with a product of k# and K,,», which cannot be separated
`empirically. One approach is to assume that k, = k*, thereby placing the
`entire difference of J* and J in the off rates. This is not likely to be a bad
`approximation, as discussedlater.
`One might ask if the expansion and linearization process represents
`mathematical overkill. For example, it might appear intuitive to assume
`that near the beginning of the reaction all variables take their original
`values, ie., c = ¢,, J = Jy, and c* = c%. Then direct integration of Eq.
`(7) gives
`
`from which
`
`Cp = J*c*t
`
`A = $I *c*t?
`
`(17)
`
`(18)
`
`If our previous expression [Eq. (12)] for A were to be expanded,this
`would be the leading term, so this simple idea is not far off. However, it
`is not at all apparent from this equation if the ¢? is a lower or upperlimit;
`as we shall see later, augmented pathway polymerization can give way to
`exponential growth after beginning with a t* time dependence.It is also
`not so clear what should be viewed as the rate constant in Eq. (18). For
`example, one might falsely conclude that the product JJ* provides the
`effective rate constant, as it has the units of reciprocal time squared, leaving
`
`®R. E. Samuel, E. D. Salmon, and R. W. Briehl, Nature 345, 833 (1990).
`
`Page 8
`
`Page 8
`
`

`

`
`
`264 [17] In Vitro PROTEIN DEPOSITION
`
`
`
`c* to provide the amplitude. That would suggest incorrectly a rate constant
`with very low concentration dependence. While the simple process of equat-
`ing of all termsto initial values without expansion is very convenient, it
`must be viewed as a kind of “quick and dirty’’ approach, rather than a
`rigorous one. The linearization approach describedearlier, although limited
`to the initial reaction, is rigorous if applied consistently. Moreover, it can
`be used in other ways, such as constructing a solution of equations with
`an activation step for which the initial values of some parameters would
`be zero.
`Two important points for the analysis of assembly must be made here.
`First, the t? dependencealoneis not that unique, and other reactions may
`lead to such a relationship. Rather it is the concentration dependences of
`the rate constants that serve to provide true diagnostics. Second, ¢? repre-
`sents the maximal time dependenceof the initial course, with further time
`“flattening” the curve. As weshall see later, otherinitial t? results can have
`the opposite effect, namely that their time dependence exceedstheinitial
`value. The reasonforthis is that in Eq. (11a) the coefficient of A is negative,
`ie., the further the reaction proceeds, the slower the rate. However, the
`opposite can happen andthis is discussed next.
`
`Polymerization with Secondary Pathway
`
`It is possible to have a term in the rate of growth of the polymer
`concentration[i.e., Eq. (11a)] that is positive so that the reaction accelerates
`rather than decelerates. Three possibilities readily come to mind: fragmen-
`tation, branching, and heterogeneous nucleation. We distinguish these in
`the following way: fragmentation is the result of breaking polymers to
`produce new polymer ends onto which growth may occur. The simplest
`model for this process is that it occurs with a rate proportionalto A,i.e.,
`that breaks are possible anywhere, and because polymers are long and
`linear, A is a good measure of the net length of all polymers. (Fancier
`models are possible in which breakage is proportional to higher orders of
`A, but that will only be manifest as higher order terms in the expansions
`discussed earlier.” Fragmentation appears to be operative for some condi-
`tion of actin filament growth.®
`Branchingis almost as simple as fragmentation and represents the begin-
`ning of a new polymer from an existing site by the addition of the first
`monomerto that site. (In other words, a polymer is not branched until a
`branch site begins the new polymer.) This is given by a term @kpyancnJe in
`
`77. L. Hill, Biophys. J. 44, 285 (1983).
`8 A. Wegner and P. Savko, Biochemistry 21, 1909 (1982).
`
`Page 9
`
`Page 9
`
`

`

`
`
`{17] 265 PROTEIN AGGREGATION KINETICS
`
`
`
`which ¢ represents the frequency of branch sites and Atranchis the rate
`constant for addition of a monomerto sucha site.
`Finally, it is possible to nucleate on the surface of a polymer. This
`resembles branchingin structural terms, but thermodynamically it is distinct
`in the same way downhill polymerization and nucleated polymerization are
`distinct, namely the new polymeris incapable of growth until a minimum
`number of monomers are present that form a heterogeneous nucleus. This
`effect is described by a term #K**/**c**, where ¢ is the fraction ofsites
`that can support heterogeneous nucleation, K** is the equilibrium constant
`for attaching a heterogeneousnucleus tothat site, J ** is the rate of elonga-
`tion of the heterogeneous nucleus, and c** is the concentration of heteroge-
`neous nuclei. Naturally, c** expected to be related to the monomer concen-
`tration by the same type relationship as Eq. (14), namely c** = Ky»c™,
`where in general the nucleus sizes n* and m* are not equal. Heterogeneous
`nucleation has been observed in sickle hemoglobin polymerization.®*°
`If we denote these various processes by Q, then Eq. (11a) becomes
`
`dc,ldt = J#c® + [Q — (dJ*c*/dc)] A
`
`with no change to Eq. (11b).
`The solution of Eqs. (11b) and (19) is then
`
`A = A(coshBt — 1)
`
`(19)
`
`(20)
`
`The cosh function begins as ¢? and then (for large Bt) becomes an exponen-
`tial, exp(Bd) (Fig. 3). (Note that it is now possible to have Br large butstill
`remain within the consistent limits for the solution. However,in the cosBt,
`the solution is limited to small Bt.)
`Now we have
`
`and
`
`Be=J jo - éUrer)|
`
`(21a)
`
`(21b)
`
`The secondary process Q has affected both the rate parameter B and the
`amplitude parameter A. Remarkably, however, the product B7A remains
`unaffected by the presence of the secondary process andisstill given by
`
`°F. A. Ferrone, J. Hofrichter, H. Sunshine, and W. A. Eaton, Biophys. J. 32, 361 (1980).
`10F, A. Ferrone, J. Hofrichter, and W. A. Eaton, J. Mol. Biol. 183, 591 (1985).
`
`Page 10
`
`Page 10
`
`

`

`
`
`266 [17] In Vitro PROTEIN DEPOSITION
`
`
`
`Concentrationofpolymerized
`
`
`
`
`
`monomers(arb.scale)
`
`0
`
`0
`
`pron
`10
`
`15
`
`5
`
`Bt
`
`Fic. 3. Polymerization kinetics with a secondary pathway. The concentration of polymerized
`monomers, A,
`is shown as a function of Bt, as given by Eq. (20). All curves are in the
`exponentiallimit of the function. A is varied from 10°? to 10° as labeled. Note thatall curves
`start from time 0, but the exponential time course gives risc to the apparent delay. A curve
`of the form A = A(Br)* is shown as the dashed line for comparison [cf. Fig. 2 and Eq. (18)].
`Contrast the delay or lag with that in Fig. 2. Only an exponential (or high power of time)
`will give the abruptness shown here.
`
`Eq. (13c) and permits a means of deducing the concentration of homoge-
`neous nuclei c* without a precise specification of the process Q.
`Thestrategy for analyzing assembly now becomes the following. When
`an abrupt time course for assembly is seen, the curve should befit to Eq.
`(20). If this succeeds, the analysis of B2A gives J*c* and the concentration
`dependence of B?A gives the nucleus size. By observing the growth of
`polymersit is possible to determine J as before. Then from either B or A
`(B being preferred) one can isolate Q and study its concentration depen-
`dence to identify which of the just-described types of secondary processit
`might be.
`Once again a word of caution is in order. The apparent lag or delayis
`the consequenceof the exponentialtime dependencesandis not a phenome-
`non of nucleation at all! For example,it is entirely possible for a downhill
`polymerization to have no nucleus,i.e., c* = c, but yet possess a secondary
`process that then gives the reaction an exponential time course and a
`distinct lag time. The lack of nucleation in that example would then be
`revealed in the concentration dependence of B7A rather than in the shape
`of the curve itself. Again, as described earlier, the lag time cannot be
`associated with a unique period during which nuclei are being formed.
`
`Page 11
`
`Page 11
`
`

`

`
`
`[17] 267 PROTEIN AGGREGATIONKINETICS
`
`
`
`Inserting Eq. (20) into Eq. (11b) reveals that the concentration of polymers
`is almost exponential, ie., most nuclei are formed after the lag!
`With a secondary process, the reaction assumes a spatial character, as
`the presence of a given polymer affects the likelihood of forming others.
`This has been explored only slightly, but the interested reader may wish
`to consult Zhou and Ferrone" or Dou and Ferrone”? for some approaches
`to this issue.
`An apparent lag time, sharper than seen in the ¢* dependence described
`earlier, is also found in cascade-type reactions, i.e., downhill polymeriza-
`tions. If there are no reverse rates it is easy to show by direct integration
`that the reaction moves forward with a power law dependence. Whatis not
`immediately obvious, but is demonstrated readily, is that the concentration
`dependenceof the characteristic rate is simply linearor, if the characteristic
`time is denoted by 7, then
`
`and
`
`so that
`
`A ~ (t/7)”
`
`tT~ Ile
`
`d log t/d log c = —1
`
`(22)
`
`(23)
`
`(24)
`
`Thus, without a secondary pathway, either the time dependence or the
`concentration dependence can be high, but not both.
`This concludes the description of basic nucleation—elongation kinetics.
`Wenowturn to a series of more detailed topics.
`
`Generalized Nucleation: Effect of Near-Nuclear Species
`
`So far our approach to describing nucleation has been based on the
`notion that a single species creates the bottleneck for growth. Whatif there
`are a few species in the pathway that have very small concentrations? This
`section provides a more rigorous way to deal with this problem following
`the treatment by Burton’? (which is based on the venerable treatment by
`Becker and Doring’*). While the exercise may appear somewhat academic,
`its importance lies in showing how the assumption of a single species in
`
`11 H.-X. Zhou and F. A. Ferrone, Biophys. J. 58, 695 (1990).
`12.Q. Dou andF.A. Ferrone, Biophys. J. 65, 2068 (1993).
`13 J. J. Burton,in “‘Nucleation Theory” (B. J. Berne, ed.). Plenum Press, New York, 1977.
`“4 R. Becker and W. Doring, Ann. Physik 24, 719 (1935).
`
`Page 12
`
`Page 12
`
`

`

`
`
`268 [17] In Vitro PROTEIN DEPOSITION
`
`
`
`the reaction path affects the final results, as well as the justification for
`ignoring the back reaction in nucleation theories.
`Weconsider the equations for the change in c;, the concentration of
`the ith species, i = 2
`
`dc;
`or = kjccj-1 + kines — (ko — kj )e;
`
`Nowdefine a flux, F;
`
`so that
`
`F,= kice; ~ kines
`
`dc;
`—=F,,-F,
`a=
`Fis
`
`Atsteady state the flux through all the states is the same,i.e.,
`
`fe kice;, — Kivicier
`
`(25)
`
`(26)
`
`27
`27)
`
`(28)
`
`The equilibrium populationswill be denoted here by uppercaseletters,1.c.,
`C;, and are such that no flux exists. In other words,
`
`Ki CC; = kins Cin
`
`Using this relationship to eliminate the back rates, we can write
`
`(29)
`
`(
`
`30
`
`)
`
`GI)
`
`f
`
`Define s(t) = c/C. Then
`
`
`
` cc; Ci+t
`
`
`
`=k?}CC;{—— -(& Cit
`
`)
`
`BoeEs
`
`CG
`
`and then if we sum over i and assumethat, past some size N, cy+1 ~ 0, we get
`
` > aoa= s(t)+ |s00 - i| Le
`
`i=1
`
`(32)
`
`Initially there are so few aggregates in the system that c ~ C, or s(t ~
`0) ~ 1, from which we get the relation
`
`fo= Is atc)
`
`i=1
`
`(33)
`
`Page 13
`
`Page 13
`
`

`

`
`
`[17] 269 PROTEIN AGGREGATIONKINETICS
`
`
`
`Another way to view this relationship is that the time for nucleation,
`(1/f,), is the sum of the times of all the preceding steps. If there is one
`species whose concentration C* is much lowerthanall others, its reciprocal
`will dominate the sum and we have
`
`fo= kiCC*
`
`(34)
`
`Becausethe flux through thesestates is the rate of formation of polymers,
`dc,/dt, Eq. (34) is essentially Eq. (7) (in which we had not distinguished
`between equilibrium values C and instantaneous steady-state valuesc).
`Note that the reverse rate has disappeared, ic., that J* = kiC. The
`validity of this was verified in an independent way by Goldstein and Stryer.?
`Equally important, it is also evident how oneis to include species of size
`nearthat of the critical nucleus,if the concentration of nuclei is not signifi-
`cantly less than the speciesof similar size.If all the monomer addition rates
`are equal, the result is particularly simple, namely
`
`fo = k&C b +|
`
`N47
`mC
`
`(35)
`
`This therefore provides a systematic way to include species of size similar
`to the nucleus. Conversely, if one assumesa single rate-limiting species, of
`concentration C*, what oneis actually determiningis an effective concentra-
`tion whose reciprocal (1/C*) is equal to the sum of the reciprocals of the
`concentrations of aggregates near the nucleus. Hence the concentration of
`nuclei in the single-species assumption underestimates the actual concentra-
`tion of nuclei; this approximation will be improved by including more
`species, as is evident from Eq. (35).
`The time required to establish the steady-state flux is also a matter of
`interest. Roughly it goes as n*7, where n* is the size of the nucleus and 7
`is the step time, approximately 1//* (see Firestoneeral.’*). Then n*z needs
`to be comparedto 1/B. In most cases, the time to establish the flux is short,
`ie., n*r < 1/B, as generally n° < c/c* where the latter inequality is based
`on the relative smallness of c*.
`
`Some Physical Issues
`
`Given the correct phenomenology, one would wish to rationalize or
`deduce from first principles the nature of the rate constants and the nucle-
`ation barrier. It becomeslogical to think in terms of an energetic barrier
`rather than in termsof the rate constants themselves as the nucleusis being
`treated as being in equilibrium with the monomers.
`
`15M. P. Firestone, S. K. Rangarajan, and R. de Levie, J. Theor. Biol. 104, 535 (1983).
`
`Page 14
`
`Page 14
`
`

`

`
`
`270 [17] In Vitro PROTEIN DEPOSITION
`
`
`
`Early approaches to nucleation imagined an abrupt transition such as
`mightoccurin the closure of a ring in the formationofhelical polymers.”*5
`This was based on the need to justify the change in the free energy by
`invoking a greater numberof contacts once the ring closed.It is important
`to ensure that such models are faithful to the assumptions of thermodynamic
`nucleation and that the contacts along the initial chain are not weaker than
`those up and down, as otherwise the structure will form in short double
`layers in preference to the single strand that later wraps arounditself. As
`pointed out earlier, it is not necessary to have a change in structure such
`as this for a nucleation barrier. If the nucleusis the result of simple thermo-
`dynamics of small clusters,its size will be a function of the initial concentra-
`tion, and the nucleation barrier will be curved rather than having a cusp
`as for a helix closure. In the case of a thermodynamic barrier, it is possible
`to construct models based on simple thermodynamic principles that reduce
`the nucleus calculation to energetic parameters. These models are beyond
`the scope of this article, but suffice it to say that the guiding principle is
`the competition between the free energy increase due to contacts within the
`nucleus versus the loss in entropy due to immobilization of the monomersin
`aggregates. In terms of entropic considerations, it is also important to
`include the redemption of entropy arising from vibration of the monomers
`themselves within the frameworkofthe aggregate.”!©-1® This is easily over-
`looked and can be very significant.
`An important feature of the thermodynamic models is that the nucleus
`size depends on the initial concentration.’” This is easy to rationalize in
`physical terms. The loss in translational entropy is the result of relative
`immobilization of the monomersin solution. The initial concentration es-
`sentially determines the volume each molecule has to “wander aroundin.”
`In a higher concentration solution, the entropy loss is not so dire as in a
`solution of lower concentration. This can easily create confusion if different
`experiments are p