A Lumry-Eyring Nucleated-Polymerization (LENP) Model of Protein Aggregation Kinetics 2. Competing Growth via Condensation- and Chain-Polymerization

Abstract

The Lumry-Eyring with nucleated-polymerization (LENP) model from part 1 (Andrews and Roberts, J. Phys. Chem. B 2007, 111, 7897 7913) is expanded to explicitly account for kinetic contributions from aggregate-aggregate condensation polymerization. Experimentally accessible quantities described by the resulting model include monomer mass fraction (m), weight-average molecular weight (M_w), and ratio of M_w to number-average molecular weight (M_n) as a function of time (t). Analysis of global model behavior illustrates ways to identify which steps in the overall aggregation process are kinetically important, based on the qualitative behavior of m, M_w, and M_w/M_n vs. t, and based on whether bulk phase separation or precipitation occurs. For cases in which all aggregates remain soluble, moment equations are provided that permit straightforward numerical regression of experimental data to give separate time scales or inverse rate coefficients for nucleation and for growth by chain and condensation polymerization. Analysis of simulated data indicates that it may be possible to neglect condensation reactions if only early-time data are considered, and also highlights difficulties in conclusively distinguishing between alternative mechanisms of condensation even when kinetics are monitored with both m and w_M.

1. Introduction

Non-native aggregation commonly refers to the process of forming protein aggregates in which the constituent monomers have significantly altered secondary structure compared to the native or folded state.^1-3 Aggregates may be soluble or insoluble, with soluble aggregates potentially ranging in size from dimers to so called high molecular weight species (~ 10 - 10³ or more monomers per aggregate).^4-6 Formation of non-native aggregates is problematic for protein based pharmaceuticals and other biotechnology products due to increased manufacturing costs, regulatory concerns, and product marketability.^3,6-8 Non-native aggregates are also implicated in a number of chronic diseases^9,10 and are suspected immunogenic agents in biopharmaceuticals.^11,12

Because non-native aggregation (hereafter referred to simply as aggregation) is typically net irreversible under the conditions that aggregates form, elucidating key mechanistic details that control aggregation kinetics is of general importance for these systems. However, even apparently simple experimental kinetics can be a convolution of multiple stages.² These may include: (partial) monomer unfolding; reversible self association or pre nucleation; nucleation of the smallest irreversible aggregates; and subsequent aggregate growth via chain polymerization and/or aggregate self association or phase separation. Furthermore, many of the kinetically relevant intermediates are often too poorly populated or transient to be directly characterized with available experimental methods.^2,6,13,14 As a result, proper deconvolution of different stages of the aggregation process requires qualitative and quantitative comparison with mechanistic mathematical models that are couched in experimentally accessible quantities such as mass-percent loss of monomer and time-dependent scattering data.^2,4,15-19

A large majority of available mathematical models for aggregation kinetics can be categorized in terms of which stage or stages in the overall aggregation process that they treat explicitly or implicitly. Currently, no available model treats all of the above stages with equally detailed descriptions for natively folded proteins. Rather, most models fall in one of two categories.² Those in the spirit of Lumry-Eyring treatments primarily consider only unfolding and folding in mechanistic detail, and use phenomenological or empirical treatments for assembly steps. Alternatively, polymerization models typically ignore conformational transitions and treat only assembly steps in detail.²

A previous report²⁰ presented a first generation Lumry-Eyring Nucleated-Polymerization (LENP) model that included thermodynamics of monomer conformational stability and prenucleation, along with dynamics of nucleation and of growth via chain polymerization. It is also one of only two models^20-22 that consider the effects of aggregate (in)solubility on experimental kinetics of monomer loss or soluble-aggregate size distributions. In this context, soluble aggregates are those that are available to consume additional protein monomers,^15-24 rather than being defined in terms of a particular size range.²⁵

The previous LENP model did not include detailed treatments of the kinetics and mechanism of aggregate-aggregate coalescence or condensation leading to soluble and/or insoluble aggregates.^2,20,22 Incorporating details of condensation is important if one is interested in quantifying the resulting aggregate size distribution, but can lead to significant added mathematical complexity.²⁶ This may explain why, for non-native protein aggregation, there are relatively few experimentally tested kinetic models that describe condensation in considerable detail, and those models have typically been system-specific.

For example, Pallitto and Murphy incorporated size-dependent, diffusion limited lateral and end to end association to describe soluble filament and insoluble fibril formation based on a priori knowledge of stoichiometry and geometry in Aβ aggregation.¹⁶ In simpler treatments, Modler et al¹⁷ and Speed et al¹⁸ considered irreversible condensation polymerization to form soluble aggregates, with rate coefficients that were assumed to be independent of polymer size (degree of polymerization). In each case, kinetic models were regressed against time-dependent measurements of one or more aspects of the aggregate size distribution, e.g., weight-average molecular weight^16-18 or z-average hydrodynamic radius.¹⁶ Condensation was determined to be an important or even dominant contribution in each case. However, in each case the models were developed for only a specific protein system, without considering global model behavior.

Furthermore, it is also common practice to fit monomer loss data to models in which condensation is inherently neglected,^15,23,27,28 even though corroborating structural evidence to support such an assumption may be available in only a fraction of reported cases.^2,6 Overall, this highlights a need for more general analysis of aggregation kinetics within a mechanistic framework that can easily distinguish which contributions are important, and that can also provide a means to quantify those contributions by regression against experimental kinetics.

The present report extends the previous LENP model to include explicit and detailed descriptions of condensation. Particular questions that are addressed include: (1) what experimental signatures easily allow one to qualitatively determine whether neglecting condensation^{20,23,24,27-29} is appropriate? (2) can one quantitatively separate contributions from condensation, chain-polymerization, and nucleation without detailed a priori knowledge¹⁶ of the association mechanism or aggregate morphology? (3) how sensitive are experimentally accessible kinetics to mechanistic details such as size-dependent vs. size-independent condensation steps? (4) how are the answers for (1) to (3) altered if one considers only early-time data (i.e., only the first few percent loss of monomer)? These questions are important for deconvoluting the effects of chemical additives or protein stabilization strategies on different stages of aggregation,^2,30-32 inferring mechanistic details of aggregate-aggregate assembly,¹⁶ and in applications such pharmaceutical product stability that typically focus on only small extents of reaction or percent loss of monomer.^3,6 Finally, this report provides the global behavior of the improved LENP model, and illustrates an application of the model to experimental data using recently reported results for aggregation of α-chymotrypsinogen A (aCgn).⁵

2. Model Description & Derivations

Table 1 summarizes key symbols and definitions used throughout this report. Fig. 1 schematically shows the six stages of nonnative aggregation that are included in the model developed and analyzed here. Stages 1 to 4 are the same as those employed in the previous LENP model.²⁰ Briefly, the six stages in Fig. 1 are: (1) conformational transitions of monomers between folded (F) and unfolded (U) states, with the possibility for stable folding intermediates (I). The monomer conformational state (e.g., F, I, or U) that is most prone or reactive with respect to aggregation is denoted R; (2) association of R monomers to form reversible prenuclei or oligomers (R_i) composed of i molecules; (3) nucleation of the smallest aggregate species that is effectively irreversible (A_x) by a conformational rearrangement step (R_x → A_x);^16,20 (4) growth of soluble aggregates via chain polymerization; (5) soluble aggregate growth due to aggregate-aggregate association such as condensation polymerization;^5,16 (6) removal of aggregates via phase separation to form macroscopic particles or precipitates.^21,33,34 In stage 6, all aggregates composed of n* or more monomers are treated as insoluble.^20-22

Table 1.

List of key symbols

Name	Definition
Aj	Agg. composed of j monomers(a)
Ax	Nucleus(a)
aj	[Aj]/C0
C0	Initial monomer concentration(a)
Cref	Std. state monomer conc.(a)
KIU	Eq. const, for I ↔ U
Ki	Eq. const, for iR ↔ Ri(b)
KNI	Eq. const, for N ↔ I
KRA	Eq. const, for Aj+R ↔ AjR(c)
ka	Monomer assoc. rate coeff.(d)
ka,x	ka for nucleation step(d)
kB	Boltzmann's constant(e)
kd	Dissociation rate coeff.(f)
kd,x	kd for Rx-1 + R ↔ Rx (f)
kg	Growth rate coeff.(d)
knuc	Nucleation rate coeff.(f)
kobs	Observed rate coeff. for monomer loss(f)
v	Apparent reaction order for monomer loss
δ	# monomers added per chain-polymerization step
kr	Rearrangement rate coeff.(f)
kr,x	kr for nucleation step(f)
ki,j	Condensation rate coeff.(d)
κi,j	ki,j/kx,x
τg	Characteristic timescale of chain polymerization
τn	Characteristic timescale of nucleation
τc	Characteristic timescale of condensation polymerization
N	Native monomer(a)
I	Intermediate state (monomer)(a)
U	Unfolded state (monomer)(a)
R	Reactive monomer(a)
fR	Fraction reactive monomer
n*	Size at which precipitation occurs
Ri	Reversible oligomer of i monomers(a)
Rx	Reversible prenucleus(a)
x	Nucleus stoichiometry
m	([N]+[I]+[U])/C0
Mnagg	Number-avg. aggregate MW(g)
Mmon	Monomer MW(g)
Mwagg	Weight-avg aggregate MW(g)
βgn	τn/τg
βcg	τg/τc
σ	Σ[Aj]/C0
λ1	1st moment of soluble agg. size distribution
λ2	2nd moment of soluble agg. size distribution
θ	Dimensionless time (t/τn)
μ	Mean of aj/σ distribution
σμ	Variance of aj/σ distribution
${\stackrel{‒}{\kappa }}_n$	Number average κi,j
${\stackrel{‒}{\kappa }}_w$	Weight average κi,j
τg(0)	τg at Cref and fR = 1
τn(0)	τn at Cref and fR = 1
τc(0)	τc at Cref

Abbreviations: agg. = aggregate, aggn. = aggregation, conc. = concentration, const. = constant, eq. = equilibrium, MW = molecular weight, unf. = unfolding

^(a)[mol/volume]

^(b)[(mol/volume)^1-i]

^(c)[(mol/volume)^-1]

^(d)[(mol/volume)^-1·time^-1]

^(e)[energy/K]

^(f)[time^-1]

^(g)[mass·mol^-1]

^(h)[Kelvin]

⁽ⁱ⁾[time]

^(j)[energy/mol]

Figure 1.

Reaction scheme with associated model parameters for the six key stages in the LENP model. The steps shown in each panel are treated as elementary irreversible (single arrow) steps, or as pre equilibrated or steady-state (double arrow) when translating them to mass action kinetic equations.

As in the previous report,²⁰ stages 1 and 2 are assumed to be fast and thus preequilibrated compared to stages 3-6. As a result, only equilibrium constants for unfolding (K_FI, etc.) and prenucleation (K_i, i = 2,…,x-1) appear in stages 1 and 2, respectively. The kinetics of conformational rearrangement as part of nucleation in stage 3 are treated by assuming a concerted, unimolecular rate-limiting step with rate coefficient k_r,x.²⁰ The balance of rearrangement (R_x → A_x) and association (R + R_x-1 → R_x) steps in stage 3 is treated with a local steady-state approximation. For association, k_a,x and k_d,x denote forward and reverse rate coefficients. Similar considerations and nomenclature are included for growth via chain polymerization (stage 4).²⁰ R monomers can reversibly self associate with pre-existing soluble aggregates, followed by a conformational rearrangement step that makes monomer addition effectively irreversible. The rate coefficients k_a, k_d, k_r and equilibrium constant K_RA in stage 4 are the same as in the earlier LENP model.²⁰ In stage 5, k_i,j denotes the rate coefficient for irreversible association of aggregates composed of i and j monomers to form a soluble aggregate of i + j monomers. Stage 6 is effectively instantaneous phase separation of any aggregate that contains n* or more monomers.

2.1 LENP model equations

The following derivations are based on the reaction scheme in Fig. 1, and employ the same nomenclature as previous work²⁰ to the extent possible here. Characteristic time scales are defined for nucleation $({\tau }_n\equiv {\tau }_n^{\left(0\right)}{f}_R^{-x}{(C_{ref}∕C_0)}^{x-1})$, growth via monomer addition $({\tau }_g\equiv {\tau }_g^{\left(0\right)}{f}_R^{-\delta }{(C_{ref}∕C_0)}^{\delta })$, and condensation $({\tau }_c\equiv \frac{{\tau }_c^{\left(0\right)}{C}_{ref}}{C_0})$ (see also Appendix). In these definitions, f_R = [R]/([N]+[I]+[U]) is the mole fraction of monomer that is in the aggregation prone conformational state. C_ref is a reference state concentration that defines the concentration scale of the standard state for association free energies and equilibrium constants. The respective intrinsic time scales (denoted with superscript (0)) are defined as ${\tau }_n^{\left(0\right)}\equiv {\left(k_{nuc}{K}_{x-1}{C}_{ref}^{x-1}\right)}^{-1}$, ${\tau }_g^{\left(0\right)}\equiv {\left(k_g{K_{RA}}^{\delta -1}{C_{ref}}^{\delta }\right)}^{-1}$, and ${\tau }_c^{\left(0\right)}\equiv {\left(k_{x,x}{C}_{ref}\right)}^{-1}$. They are termed intrinsic because they are independent of initial monomer concentration and the free energy of monomer conformational transitions. k_g ≡ k_ak_r/(k_d + k_r) is the effective rate coefficient for chain polymerization, and k_nuc ≡ k_a,xk_r,x/(k_d,x + k_r,x) is that for nucleation.²⁰

The above definitions along with the derivations elsewhere²⁰ and in the Appendix show that although there are numerous parameters in Fig. 1 and Table 1, the assumptions of preequilibration for stages 1 and 2, and local steady state for stages 3 and 4 reduce the total to only seven distinguishable parameters or functions: τ_n and x account for stages 1, 2, and 3; τ_g and δ account for stage 4; and n* accounts for stage 6. Stage 5 is accounted for by τ_c and κ_i,j ≡ k_i,jC₀τ_c. κ_i,j may be a function of i and j, but its (i,j) dependence is uniquely set by the choice of mechanistic model describing size-dependent condensation (see also below and Sec. 2.3). Therefore, there are six adjustable model parameters once the condensation mechanism is selected.

The Appendix provides additional details regarding derivations of the kinetic working equations for monomer and all soluble aggregates. Eqs. A1, A4, and A5 are the dynamic material balances based on Figure 1 and mass action kinetics. They can be rewritten in nondimensional form by defining θ = t/τ_n, β_gn = τ_n/τ_g, and β_cg = τ_g/τ_c to give

$$\frac{dm}{d\theta }=-x{m}^x-\delta {\beta }_{gn}{m}^{\delta }\sigma$$ (1)

$$\frac{d{a}_x}{d\theta }=m^x-{\beta }_{gn}{a}_x{m}^{\delta }-{\beta }_{cg}{\beta }_{gn}{\kappa }_{x,x}{a_x}^2-{\beta }_{cg}{\beta }_{gn}{a}_x\sum _{j=x}^{n\ast -1}{\kappa }_{x,j}{a}_j$$ (2)

$${\phantom{\mid }\frac{d{a}_i}{d\theta }\mid }_{\underset{i\in \left\{\mathit{even}\right\}}{x<i<n\ast }}={\beta }_{gn}(a_{i-1}-a_i)m^{\delta }-{\beta }_{cg}{\beta }_{gn}{\kappa }_{i,i}{a_i}^2-{\beta }_{cg}{\beta }_{gn}{a}_i\sum _{j=x}^{n\ast -1}{\kappa }_{i,j}{a}_j+{\beta }_{cg}{\beta }_{gn}\sum _{j=x}^{i∕2}{\kappa }_{i-j,j}{a}_{i-j}{a}_j$$ (3)

When i in Eq. 3 is odd, the right-most summation runs from x to (i-1)/2 instead of i/2. The dimensionless monomer concentration is m ≡ ([N]+[I]+[U])/C₀, with contributions from [R_i] neglected for K_iC₀<<1;²⁰ dimensionless concentrations for nuclei (a_x ≡ [A_x]/C₀) and larger irreversible aggregates (a_i ≡ [A_i]/C₀) are similarly defined. The dimensionless condensation rate coefficient is defined as κ_i,j (≡ k_i,j/k_x,x). If there is no size dependence for condensation, κ_i,j = 1 is a constant for all (i, j) pairs. The model parameters that determine the characteristic behavior of the solutions to Eq. 1-3 are [x,δ,κ_i,j,β_gn,β_cg,n*]. Eq. 1 is identical to the previous version of the LENP model.²⁰ Eq. 2-3 are more complex than in the earlier model, as they include contributions from condensation (i.e., the terms in which β_cg appears). If one neglects condensation (β_cg = 0, τ_c → ∞), Eq. 2-3 are equivalent to the condensation free model in ref. 20.

In general, Eqs. 1-3 cannot be solved exactly in analytical form. They can be numerically integrated to simulate the time profiles for monomer concentration on a mass fraction basis (m), as well as the size distribution of aggregates and all associated moments of that distribution. The former quantity is often experimentally accessible by techniques such as size exclusion chromatography (SEC) and field flow fractionation (FFF).^13,14,35 Indirect measures of m might also be useful, provided they can be properly calibrated against direct measurements.⁶ Examples include dye binding,^36,37 changes in beta sheet content monitored spectroscopically,^15,38,39 and turbidity or optical density (provided all aggregates are insoluble).³⁴ In contrast, the detailed or precise size distribution (a_j vs. j) is not usually accessible experimentally. However, techniques employing static or dynamic laser light scattering are able to provide exact or approximate values for the weight-average molecular weight (M_w) and the ratio of M_w to the number-average molecular weight (M_n). The ratio M_wM_n is the polydispersity of the size distribution.⁴⁰ Using techniques such as SEC or FFF with in-line light scattering detection,^5,13,35,41 it also possible to separately measure the weight-average molecular weight of the aggregate size distribution ($M_w^{agg}$), and to provide a lower bound on the polydispersity of that distribution, $M_w^{agg}∕M_n^{agg}$ ⁵

Using the nomenclature here, the weight- and number-average molecular weights of soluble aggregates are

$$\frac{M_w^{\mathit{agg}}}{M_{\mathit{mon}}}=\frac{\sum _{j=x}^{n\ast -1}{j}^2{a}_j}{\sum _{j=x}^{n\ast -1}j{a}_j}=\frac{{\lambda }_2}{{\lambda }_1}=\frac{{\lambda }_2}{1-m}$$ (4a)

$$\frac{{M_n}^{\mathit{agg}}}{M_{\mathit{mon}}}=\frac{\sum _{j=x}^{n\ast -1}j{a}_j}{\sum _{j=x}^{n\ast -1}{a}_j}=\frac{{\lambda }_1}{\sigma }$$ (4b)

with M_mon denoting the molecular weight of a monomer, and the superscript agg indicating that the summations are carried out over all soluble species that do not assay as monomers. For the present case, this makes the lower bound j=x in the summations in Eq. 4. This is expected under conditions where prenuclei are thermodynamically disfavored (low values of K_iC₀).²⁰ Equivalent expressions can be derived if one can experimentally resolve smaller aggregates or if it is not convenient to separate monomer contributions in the assay being employed.^16-18,20

Eq. 4 also relates $M_w^{agg}$ and $M_n^{agg}$ to the first and second moments of the soluble aggregate size distribution (λ₁ and λ₂, respectively).

$${\lambda }_1\left(t\right)=\sum _{j=x}^{n\ast -1}j{a}_j\left(t\right)$$ (5a)

$${\lambda }_2\left(t\right)=\sum _{j=x}^{n\ast -1}{j}^2{a}_j\left(t\right)$$ (5b)

For n* → ∞, λ₁ is equal to the fractional monomer loss (1-m) at a given time. The zeroth moment of the aggregate size distribution is equal to σ as it was defined previously²⁰ (see also Appendix). Physically, σ is the total number of aggregates per unit volume, scaled by the initial protein concentration on a monomer basis. These moments are not normalized (e.g., σ is not 1). It follows from Eq. 4-5 that the polydispersity of the aggregate size distribution can be expressed as

$$\frac{{M_w}^{\mathit{agg}}}{{M_n}^{\mathit{agg}}}=\frac{{\lambda }_2\sigma }{{{\lambda }_1}^2}$$ (6)

2.2 Moment Equations for Soluble Aggregate Conditions

For cases where aggregates remain soluble throughout the course of an experiment (i.e., large n*), Eqs. 1-3 present an essentially infinite set of coupled, non linear ODEs. These must be repeatedly solved numerically to regress model parameters from experimental data unless one instead employs approximate, analytical solutions. Examples of accurate analytical solutions when condensation can be neglected were the focus of ref. 20 and have been previously used for regression against experimental data.^4,15,19,42 However, those treatments do not provide a means to describe changes in the aggregate size distribution when condensation is appreciable.²⁰ An alternative approach is to replace Eqs. 1-3 by summing across all aggregate sizes (j) to provide differential equations for the time dependence of m and different moments of the aggregate size distribution (see also Appendix). Under conditions where nucleation is slow compared to aggregate growth, Eqs. A4-A6 are accurate approximations, and with Eq. A1 lead to

$$\frac{dm}{dt}=-x\frac{m^x}{{\tau }_n}-\frac{\delta }{{\tau }_g}{m}^{\delta }\sigma$$ (7)

$$\frac{d\sigma }{dt}=\frac{m^x}{{\tau }_n}-\frac{{\stackrel{‒}{\kappa }}_n}{2{\tau }_c}{\sigma }^2$$ (8)

$$\frac{d{\lambda }_2}{dt}=x^2\frac{m^x}{{\tau }_n}+\frac{1}{{\tau }_g}{m}^{\delta }({\delta }^2\sigma +2\delta (1-m))+\frac{{\stackrel{‒}{\kappa }}_w}{{\tau }_c}{(1-m)}^2$$ (9)

with

$${\stackrel{‒}{\kappa }}_n\equiv \frac{\sum _{i=x}^{\infty }\sum _{j=x}^{\infty }{\kappa }_{i,j}{a}_i{a}_j}{\sum _{i=x}^{\infty }\sum _{j=x}^{\infty }{a}_i{a}_j}$$ (10a)

$${\stackrel{‒}{\kappa }}_w\equiv \frac{\sum _{i=x}^{\infty }\sum _{j=x}^{\infty }{\kappa }_{i,j}\left(i{a}_i\right)\left(j{a}_j\right)}{\sum _{i=x}^{\infty }\sum _{j=x}^{\infty }\left(i{a}_i\right)\left(j{a}_j\right)}$$ (10b)

and with the first moment (λ₁ replaced by m. Eq. 10 defines number-average and weight-average κ_i,j values (${\stackrel{‒}{\kappa }}_n$ and ${\stackrel{‒}{\kappa }}_w$, respectively). In the most general case, ${\stackrel{‒}{\kappa }}_n$ and ${\stackrel{‒}{\kappa }}_w$ are not constant because they depend on the aggregate distribution {a_j}, and this distribution changes as aggregation proceeds. The simplest case mathematically is with ${\stackrel{‒}{\kappa }}_n$ and ${\stackrel{‒}{\kappa }}_w$ identical to unity at all times, and occurs when κ_i,j is independent of size.

Eq. 7-10, along with Eq. 4 provide a numerically tractable means for parameter estimation based on experimental kinetic data for monomer loss and aggregate molecular weight. However they are applicable only when all aggregates remain soluble. If appreciable aggregate phase separation (precipitation) occurs, Eq. 1-3 or simplified limiting cases such as shown below (Sec. 3) and elsewhere²⁰ must instead be used.

2.3 A Simple Size-Dependent Condensation Model

As a test case to explore the effects of a physically plausible size dependence for κ_i,j, a difffusion-limited Smoluchowski model^43,44 for aggregate association rates was selected (see also, Sec. 3.2).

$$k_{i,j}=4\pi N_A(D_i+D_j)(R_i+R_j)f$$ (11)

N_A is Avogadro's number; D_i and D_j are the translational diffusion coefficients for aggregates composed of i and j monomers, respectively; R_i and R_j are the respective contact radii; and f is a steric factor that accounts for the fact that only a fraction of the surface of the aggregate(s) may be “reactive” with respect to contacting another aggregate. For simplicity, f was assumed to be independent of i and j, and the Stokes Einstein equation was applied for the translational diffusion coefficients. The resulting expression is

$$k_{i,j}=\frac{2N_A{k}_{B}T}{3\eta }(\frac{1}{R_i}+\frac{1}{R_j})(R_i+R_j)$$ (12)

where k_B is Boltzmann's constant, T is the absolute temperature, and η is the viscosity of the solvent. Analogous but more complex expressions can be derived by assuming different aggregate morphologies and/or details of the aggregate-aggregate association process.^16,45 Using Eq. 12 in the definition of κ_i,j, and making the simplifying approximation that R_j ~ j gives

$${\kappa }_{i,j}=0.25(i^{-1}+j^{-1})(i+j)$$ (13)

${\stackrel{‒}{\kappa }}_n$ and ${\stackrel{‒}{\kappa }}_w$ are calculated based on the time-dependent aggregate size distribution {a_j} as it is updated during numerical integration (see also Eq. 10). It is not possible to solve Eq. 7-10 with a size-dependent κ_i,j unless one assumes or knows the relationship between {a_j} and the moments of the distribution. For illustration purposes here, simple discrete probability distribution functions (pdf) were used to describe the aggregate size distribution with mean (μ) and variance (σ_μ):

$$\mu =\frac{\sum _{i=x}^{\infty }i{a}_i}{\sum _{i=x}^{\infty }{a}_i}=\frac{{\lambda }_1}{\sigma }=\frac{M_n^{\mathit{agg}}}{M_{\mathit{mon}}}$$ (14a)

$${\sigma }_{\mu }=\frac{\sum _{i=x}^{\infty }{i}^2{a}_i}{\sum _{i=x}^{\infty }{a}_i}-{\left(\frac{\sum _{i=2}^{n\ast }i{a}_i}{\sum _{i=2}^{n\ast }{a}_i}\right)}^2=\frac{{\lambda }_2}{\sigma }-{\left(\frac{{\lambda }_1}{\sigma }\right)}^2$$ (14b)

For under dispersed distributions (σ_μ < μ) the bionomial pdf was used, while for equal or overdispersed distributions (σ_μ ≥ μ) the negative bionomial pdf was used.^46,47 In each case, the (normalized) pdf is completely specified by the mean and the variance, and these in turn are set by σ, λ₁ (or m) and λ₂.

Alternative models for the size dependence of κ_i,j and for pdfs to approximate the aggregate size distribution were also considered. However, a systematic study of each was foregone, as there are many possible alternatives and the purpose of considering a size-dependent κ_i,j in the present study was only to qualitatively assess the utility and limitations of using the simpler, size-independent κ_i,j approximation that is commonly used.^17,18,21

3. Results & Discussion

Solutions to the LENP model (Eqs. 1-5) were simulated systematically over a wide range of model parameters, including: x = 2-10, β_gn = 10^-1 -10³, β_cg = 0-10³, and n* = 10 to 2×10⁴ (effectively n*→∞). The value of δ was set as 1 for all simulated results reported below. Results for δ >1 were tested for selected conditions, and all derivations and resulting working equations below do not require δ=1 to be assumed. Additional parameter values beyond the extremes of the ranges listed above were also tested to confirm that no qualitative changes in behavior were observed by extending the parameter ranges. The initial conditions in each case were m = 1, σ = 0, a_j = 0 (x ≤ j < n*).

Four main outputs from the model solutions are (each as a function of time): (1) monomer loss kinetics on a mass fraction basis, m(t) and dm/dt; (2) the zeroth moment or total number concentration of the aggregate size distribution, σ(t) and dσ/dt; (3) weight-average molecular weight of soluble aggregates, $M_w^{agg}∕M_{mon}$; (4) aggregate polydispersity, $M_w^{agg}∕M_n^{agg}$. As noted in Sec. 2.1, outputs (1), (3), and (4) are directly or indirectly accessible in in vitro experiments. Typically, σ(t) is not directly accessible via experiment, but its behavior is a useful indicator of qualitatively distinct kinetic regimes ²⁰ (see also below).

3.1 Global Model Behavior

Numerical solutions to the LENP model (Eq. 1-5) across a broad range of model parameter values with κ_i,j = 1 displayed qualitatively distinct regimes or types of behavior in terms of experimental observables. Table 2 summarizes the different types or categories of limiting behavior, using nomenclature based on previous reports.^20,21 The type of qualitative behavior the model exhibits is dictated mathematically by the values of the five key dimensionless groups or parameters noted above (n*, x, δ, β_gn, β_cg). Figures 2 and 3 illustrate the qualitative behaviors in terms of m(t), σ(t), $M_w^{agg}∕M_{mon}\mathrm{vs}.(1-m)$, and $M_w^{agg}∕M_n^{agg}\mathrm{vs}.(1-m)$. Each of these quantities except σ can be experimentally determined quantitatively or semi quantitatively. The behavior of σ is included because it provides insight into the behavior of m(t) in each case. In Figures 2 and 3, t is scaled by t₅₀ in order to more easily compare profiles with greatly different absolute time scales; t₅₀ is defined by m(t = t₅₀) = 0.5.

Table 2.

Summary of key experimental signatures and scaling behaviors for each kinetic type produced by the LENP model. Examples of illustrative profiles are given in Figures 2 and 3. Expanded from Ref. 20

Kinetic Type	γ(a)	v(b)	α(c)	Polydispersity	$M_w^{\mathit{agg}}∕M_{\mathit{mono}}$
Ia
Low βgn	x -1	x	x	High	increases strongly and non-linearly with (1-m)
High βgn	x /2	x /2+1	x /2+1	High
Ib	x -1	x	x	Insoluble Agg. / Precipitates Present
Ic	x -1	x	x	Low	ca. constant (= x) versus (1-m)
Id	Similar to Ic with small increase of polydispersity and $M_w^{\mathit{agg}}$
II	(x + δ -1)/2	δ	(x + δ)/2	Low	linear in (1-m)
III(e)	x -1	x, 1	x, 1	Insoluble Agg. / Precipitates Present
IVa	Early time (< ca. t50) similar to Ic (low βgn) or II (high βgn)
IVb	Early time (< ca. t50) similar to II; precipitates present at longer times.

Type	Physical Scenario
Ia	Low βgn: nucleation-controlled monomer loss, condensation-dominated growth
	High βgn: slow nucleation, competing condensation and chain polymerization
Ib	polymerization to low-Mw, soluble aggregates with immediate precipitation
Ic	nucleation only; negligible chain polymerization, condensation, or precipitation
Id	nucleation with slow chain polymerization; negligible condensation or precipitation
II	slow nucleation, fast chain polymerization; negligible condensation or precipitation
III	type Ib with aggregates saturated at (finite) solubility limits
IVa	slow nucleation; condensation appreciable after t > ca. t50
IVb	slow nucleation; fast polymerization; precipitation appreciable after t > ca. t50

^(a)γ defined by 1/t₅₀ ~ k_obs ~ (C₀)^γ

^(b)v defined by dm/dt = -k_obsm^v over ca. one or more half lives

^(c)α defined by k_obs ~ f_R^α

^(e)Ref. 21

Figure 2.

Illustrative profiles of limiting behaviors produced by the LENP model under conditions of fast chain polymerization relative to nucleation, based on simulations of Eq. 1-5 with β_gn = 1000, x = 6, δ = 1, and ${\stackrel{‒}{\kappa }}_n={\stackrel{‒}{\kappa }}_w\equiv 1$ for different regimes of n* and β_cg. Qualitatively distinct behaviors are labeled according to the text in Section 3. Types Ia (solid gray), IVa (dash black), and II (both solid black and dotted gray) correspond to β_cg = 10, 0.5, 0.05 and 0, respectively, with n* → ∞. Type Ib (dotted black) corresponds to β_cg = 10 and n* = 10. The panels show (A) monomer loss kinetics, (B) $M_w^{agg}$ as a function of the extent of reaction (1-m), (C) dimensionless number concentration (σ) of soluble aggregates available for further growth; (D) polydispersity of the aggregate size distribution as a function of (1-m). Polydispersity of type Ib is not shown, as experimental polydispersity results would be convoluted by the presence of insoluble/precipitating particles under type Ib conditions.

Figure 3.

Analogous profiles to those in Figure 2, but under conditions of slow chain polymerization compared to nucleation; β_gn = 0.1, other parameters are the same as in Fig. 2. Types Ia (solid gray), IVa (dash black), and Ic (both solid black and dotted gray) correspond to β_cg at 1000, 50, and 0.01 and 0, respectively.

In Table 2, the scaling exponents correspond to limiting behaviors of the effective or observed rate coefficient for monomer loss (k_obs) and apparent reaction order (v), defined by

$$\frac{dm}{dt}\cong -k_{obs}{m}^v$$ (15)

when m(t) is considered over multiple half lives.²⁰. The scaling relationships were derived previously²⁰ for most entries in Table 2, and are included here for completeness when the new features are presented below. The primary new results are for the behavior of $M_w^{agg}∕M_{mon}$ when comparing conditions where condensation is negligible or appreciable. The key features of types Ia, Ib, Ic, II, and IVa/IVb are briefly reviewed below. Type III occurs only if aggregate solubility limits are reasonably large,²¹ and is not reviewed further here. For reference, Figure 4 provides illustrative state diagrams that show ranges of model parameter values over which each kinetic type occurs. Each choice of parameter values for simulated profiles in Fig. 2 and 3 correspond to a state point in Figure 4.

Figure 4.

Kinetic state diagrams illustrating the placement of types Ia, Ib, Ic, II, and IVa/b within the space of model parameter values (x = 6, δ = 1, ${\stackrel{‒}{\kappa }}_n={\stackrel{‒}{\kappa }}_w\equiv 1$ for all points). Panel A: varying β_gn and β_cg at fixed n* → ∞. Panel B: varying β_cg and n* at fixed β_gn = 1000. n* → ∞ (denoted as `inf') is also included for comparison; Different symbols denote different limiting case behaviors illustrated in Fig. 2, 3: Ia (open triangle), Ib (filled triangle), Ic (filled diamond), II (open circle). Types IVa/b (blank space) are intermediate behaviors that bridge different limiting cases (see also discussion in text).

Type Ia denotes cases in which high molecular weight soluble aggregates form via a combination of nucleated-chain polymerization and condensation polymerization, and the rates of condensation are similar to or much greater than those for chain polymerization. Characteristic features of type Ia kinetics include: all aggregates remain soluble; v ≥ 2 (Figs. 2A, 3A), k_obs scaling with C₀ to at least the first power, $M_w^{agg}∕M_{mon}$ increasing as (1-m) raised to a power much greater than 1 (Figs 2B, 3B), and high polydispersity values (Figs. 2D, 3D). The relationships between the scaling parameters for type Ia depend on whether chain polymerization slow or fast compared to nucleation (low or high β_gn, respectively). In either case, σ shows a rapid initial increase, but declines rapidly before m declines much below 1 (Figs. 2C, 3C). This occurs because condensation rapidly decreases the number concentration of aggregates, as each condensation step consumes two aggregates (a_i and a_j) but produces only one (a_i+j). In terms of global model behavior (Figure 4), type Ia occurs for n*→∞, and high β_cg values for a given value of β_gn. The approximate locations of boundaries between different types on the state diagrams are only weakly dependent on x and δ (not shown).

Type Ib denotes cases in which all aggregates that form are either insoluble (low n*) or soluble aggregates grow so rapidly to n* that they are present at levels that are too low to be easily detectable. Characteristic features of type Ib kinetics include: visible precipitates present at low extents of reaction (m near 1); v = x ≥ 2 (Figs. 2A, 3A) and k_obs ~ C₀^x-1; and essentially undetectably low soluble aggregate concentrations (Fig. 2C). Little or no information regarding $M_w^{agg}∕M_{mon}$ or polydispersity is accessible because of the low total soluble aggregate concentrations. In terms of global model behavior (Figure 4), type Ib occurs for low n*, or for larger finite n* values when values of β_cg and/or β_gn are large.

Type Ic denotes cases in which soluble aggregates nucleate but do not phase separate or grow to much larger sizes on the time scale of monomer loss. Characteristic features of type Ic kinetics include: all aggregates remain soluble; v = x ≥ 2 (Figs. 2A, 3A) and k_obs ~ C₀^x-1; low values of $M_w^{agg}∕M_{mon}$, and $M_w^{agg}∕M_{mon}$ (Figs. 2D, 3D). σ increases monotonically to a relatively large plateau value (Figs. 2C, 3C) because aggregates do not grow by condensation and do not reach solubility limits. In terms of global model behavior (Figure 4), type Ic occurs for low n* or high n*, provided that β_cg and β_gn are both small.

Type II denotes cases in which soluble aggregates nucleate and then grow predominantly via chain polymerization. Characteristic features of type II kinetics include: all aggregates remain soluble; v = δ ≥ 1 (Figs. 2A, 3A) and k_obs ~ C₀^(x+†-1)/2; $M_w^{agg}∕M_{mon}$ scales linearly with (1-m) once m is significantly less than 1 (Figs. 2B, 3B, and discussion below); low polydispersity that depends only weakly on extent of reaction (Figs. 2D, 3D) σ increases monotonically to a plateau value (Figs. 2C, 3C) because aggregates do not grow by condensation and do not reach solubility limits. The plateau value is relatively low because chain polymerization is fast compared to nucleation, and therefore only a small number of nuclei form before the monomer pool is depleted due to chain polymerization.. In terms of global model behavior (Figure 4), type II occurs for high n*, with low β_cg and high β_gn.

When all aggregates remain soluble (limit of large n*), $M_w^{agg}$ can be formally expressed as

$$M_w^{\mathit{agg}}=\frac{M_w^{\mathit{agg}}}{M_n^{\mathit{agg}}}{M}_n^{\mathit{agg}}=\left(\frac{M_w^{\mathit{agg}}}{M_n^{\mathit{agg}}}\right)\frac{1-m}{\sigma }$$ (16)

The second equality in Eq. 16 follows from Eq. 4b and the identity λ₁ = (1-m) for large n*. Eq. 16 shows that $M_w^{agg}$ is linear in (1-m) with a positive, non-zero slope for conditions where the polydispersity ($M_w^{agg}∕M_n^{agg}$) and the number concentration of aggregates (σ) do not change appreciably as monomers are consumed. Physically, this is the case for type II kinetics as summarized above. Analogous but less general relationships were derived phenomenologically in ref. 20. Eq. 16 also applies for types Ia and Ic, and shows the mathematical basis for the scaling behavior of $M_w^{agg}$ with (1-m) summarized above and Table 2. Eq. 16 is not valid for type Ib because $M_n^{agg}$ is not equal to (1-m) once insoluble aggregates form.

Type IV (a or b) behavior is effectively an intermediate or transition between limitingcase behaviors (types Ia, Ib, Ic, II). As such, type IV behavior has some features in common with each of the limiting case behaviors that bound it in the model parameter space. Type IV is included for the sake of completeness in Table 2 and Figures 2 - 4. The subcategories (IVa vs. IVb) distinguish which limiting-case behaviors bound the type IV transition region in the state diagrams below. For reference, type IVb is mathematically equivalent to what was simply termed type IV behavior in the earlier LENP model.²⁰

Aggregation kinetics for Aβ,¹⁶ phosphoglycerate kinase,¹⁷ and P22 tailspike¹⁸ have each been successfully described by models that are simlar to or formally the same as type Ia. α-chymotrypsinogen A displays type II behavior under sufficiently acidic (pH < ca. 4) and low ionic strength buffer conditions,^5,15,19 but shifts to Ia behavior at higher ionic strengths,⁵ or shifts to Ib behavior at pH closer to neutral (unpublished results). Illustrative data for aCgn are included explicitly in Sec. 3.3.

A number of experimental systems qualitatively behave like type Ib or IVb (or III, see ref. 39), in that aggregates precipitate,^26,34,42,48 but to best of our knowledge only bG-CSF has been explicitly modeled as type Ib or III, and shown to exhibit the quantitative scaling behaviors listed in Table 2.²¹ Unfortunately, it is often the case that published reports do not explicitly indicate whether and/or when precipitation was observed during the course of measurements of m(t). Therefore it is difficult to determine whether additional systems may be well-described by the LENP model with finite n*. To the best of our knowledge, no previous models other than the direct precursors to this work^20-22 explicitly account for the effects of aggregate insolubility on monomer loss kinetics or soluble aggregate size distributions.

Finally, Eq. 1-3 allow simulation of the complete aggregate size distribution, as shown in Fig. 5A-B (conditions same as in Fig. 2). Evolution of the aggregate size distribution as monomer loss progresses is illustrated for type II (β_cg = 0, Fig. 5A) and type Ia (β_cg = 10, Fig. 5B). As expected, condensation results in a broader size distribution and decreased total number of aggregates. If one uses only moment based kinetic equations (e.g., Eq. 7-10 in the present case), it is necessary to assume the relationship between the aggregate size distribution and the particular moments. Sec. 2 described a simple way to estimate the aggregate size distributions from moment based simulations including only zeroth, first, and second moments. Fig. 5C shows that the resulting size distributions are semi quantitatively in agreement with those from the full model (Eq. 1-3, Fig. 5A) under conditions where condensation is negligible. Comparing Fig. 5D with Fig. 5B shows that the moment-based simulations correctly predict that the distribution greatly broadens with time when condensation is appreciable. However, there are qualitative differences in the shape of the distributions from the full model that cannot be captured without assuming a more complex form for the underlying distributions. This highlights a potential limitation of moment-based models if only a limited number of moments are experimentally accessible (see also discussion below).

Figure 5.

Illustrative size distributions of soluble aggregates as a function of the extent of monomer conversion, based on simulations with x = 6, δ = 1, β_gn = 1000, and n* → ∞. Panels A (β_cg = 0) and B (β_cg = 10) correspond to simulations with Eqs 1-3 and ${\stackrel{‒}{\kappa }}_n={\stackrel{‒}{\kappa }}_w\equiv 1$. Panels C (β = 10^-4) and D (β_cg = 20) correspond to simulations with Eq. 7-10 and size dependent condensation rate coefficients (Eq. 10-14). Labels indicate the monomer mass fraction remaining for a given curve.

3.2 Parameter Estimation with the LENP model

For aggregates that remain soluble and are able to grow rapidly compared to nucleation, Eqs. 7-10 combined with Eq. 4 provide a computationally simple means to quantify separate characteristic time scales of nucleation, chain polymerization, and condensation using data regression against m(t) and $M_w^{agg}\left(t\right)$ simultaneously.

To assess the accuracy of τ_n, τ_g, and τ_c values regressed with moment equations, simulated m(t) and $M_w^{agg}\left(t\right)$ data over a common time range (4×t₅₀) were generated using Eqs. 1-5 with β_gn = 1000, x =6, δ =1, and κ_i,j ≡ 1 (κ_n = κ_w = 1), with β_cg systematically increased from 0 to 10. Only data points at selected time intervals were used for regression, so as imitate typical experimental data without in situ measurements. The results below do not change substantially if a larger number and finer spacing of data points are used. The simulated data sets were nonlinearly regressed against Eqs. 7-9 and Eq. 4.

As a test of whether models that neglect condensation can reasonably fit data in which condensation is appreciable, The same simulated data were also regressed with τ_c → ∞; Therefore, fitting only τ_g and τ_n. Furthermore, simulated data sets from Eq. 1-5 were truncated at successively smaller extents of reaction (i.e., “early time” data only), and regression vs. Eq. 7-9 was repeated. The latter two cases help to address the question of whether {m, M_w^agg} data can reliably differentiate between aggregation models that do not include condensation steps, depending on whether one uses data over multiple half lives^5,16-20 or only under early-time conditions.^23,28,49

Figure 6 compares The regressed time constant values (τ_i,fit, i = g, n, c) to the true values (τ_i,true, i = g, n, c) for The cases described above. The 95% confidence intervals of the fitted parameters and coefficient of determination (R²) are included in Fig. 6 to illustrate the quality of the fit in each case. The size and distribution of residuals were also examined to evaluate the quality of each fit (not shown), and were found to be consistent with the magnitude of confidence intervals and R² values reported below. The model parameters x and δ are necessarily integers in the LENP model, and so were held constant to avoid unnecessary complications of working with mixed-integer regression. Instead, The values of x and δ were systematically varied over physically plausible ranges (x ≥ 2, δ ≥ 1) and regression of τ_n, τ_g, τ_c was repeated for each pair of x and δ values.

Figure 6.

Comparison of values for τ_g (gray), τ_n (white), and τ_c (black) obtained by regression of Eq. 7-9 against simulated experimental data from Eq. 1-3 (see text for additional details). (A) ${\stackrel{‒}{\kappa }}_n={\stackrel{‒}{\kappa }}_w\equiv 1$ in both simulated data and fits; simulated data span four half lives. (B) same as A, but fits assumed τ_c → ∞ to imitate condensation free models. (C) same as B, but with simulated data sets truncated at the extent of reaction indicated by the label beside each set of bars. Error bars represent 95% confidence intervals from nonlinear least squares fits.

The best-fit results in Fig. 6 are for δ = 1, as all other δ values produced clearly inferior fits (not shown). However, fits with different values of nucleus stoichiometry (x) were not statistically distinguishable unless very large x values (> ca. 10) were used. The large-x fits were clearly inferior to the small-x fits, but it was not possible to further distinguish a best-fit x value. This is not unexpected based on previous analysis that showed reliable determination of x values required kinetic data over a relatively wide range of initial protein concentrations (C₀).²⁰ For concreteness, the results in Fig. 6 are for x = 6, the same value of x used to generate the simulated data from Eq. 1-3. More generally, this result highlights inherent difficulties in determining nucleus size from data regression vs. kinetic models when the data are available at only one or a small range of C₀ values.

The results in Fig. 6A show that regression against Eq. 7-9 provides accurate parameter values for a given set of m(t) and $M_w^{agg}\left(t\right)$ data. This includes conditions where condensation is negligible (β_cg << 1) and where it is the dominant mode of growth (β_cg >> 1). In all cases, the accuracy of fitted parameters was within 5% of the true values, R² values were greater than 0.99, and residuals were small and evenly distributed. In contrast, Fig. 6B shows that fitting with a model in which condensation is neglected clearly produced poor fits and inaccurate fitted parameter values under conditions where condensation is appreciable (β_cg ~ 1) or dominant (β_cg >> 1).

Figure 6C illustrates instead that if one is able to consider sufficiently early-time conditions (m ® 1), it is possible to obtain reasonably accurate values of τ_g and τ_n with a model that neglects condensation. No values of τ_c are shown because τ_c ® ∞ for the fits in Fig. 6C. The labels above each data set in Fig. 6C indicate the value of m at which the data were truncated for fitting. The truncation m value for a given data set was selected as the point at which the polydispersity first rose above a threshold value of $M_w^{agg}∕M_n^{agg}=1.1$ (cf. Fig. 2D and discussion below). The results in Fig. 6C are perhaps not surprising because the initial conditions considered here are ones in which aggregates are not present, and because condensation rates are proportional to the square of the total aggregate concentration (i.e, σ²) while chain polymerization rates are linear in σ. Thus, condensation rates do not become appreciable until larger amounts of monomer have been consumed to create new aggregates. One can reach the same conclusion via an analytical perturbation solution (results not shown), such as applied previously to a condensation-free model.²³ The above arguments notwithstanding, even with early-time data it is not possible to deconvolute τ_g and τ_n unless both m(t) and $M_w^{agg}\left(t\right)$ data are employed.

In practical terms, it is unlikely that one will know a priori whether experimental data are collected for sufficiently early times to assure condensation can be neglected. The results in Fig. 6C, when compared to those in Fig. 2D, support the empirical practice of considering condensation to be negligible if the sample polydispersity remains relatively low $(M_w^{agg}∕M_n^{agg}~1.1-1.2)$.^4,5 The results in Fig. 2C suggest an additional criterion for neglecting condensation is that M_w^agg scales linearly with (1-m). Ideally, however, it seems most prudent to instead consider models that include growth via both monomer addition and aggregate-aggregate condensation when attempting to regress accurate and mechanistically sound parameter values from experimental kinetics. An example of this approach applied to experimental data for aCgn aggregation is provided below (Sec. 3.3).

For simplicity, all preceding examples in this section used only the case of size-independent rate coefficients for condensation (κ_i,j = 1). From a practical standpoint, it also is often convenient to assume size-independent condensation so as to reduce the computational burden and complexity of models for regression.^17,18 Furthermore, it is not clear a priori that typical experimental kinetic measurements provide sufficient information to reliably distinguish between different condensation-mediated growth mechanisms. This motivates the question, can experimental m(t) and $M_w^{agg}\left(t\right)$ data robustly distinguish between different models for condensation-mediated growth?

In order to address this question, Eq. 7-10 were solved with a simple diffusion-limited Smoluchowski model for κ_i,j (cf., Section 2) to provide simulated kinetic data that were then regressed against Eq. 7-9 with the size-independent condensation model used above. Illustrative results are shown here for simulated data (size-dependent κ_i,j) with β_gn = 1000, β_cg = 1,10,20. Figure 7A shows results for β_cg = 20. The size-independent model provided excellent fits to size-dependent simulated data in all cases, with R² > 0.99 and small, evenly distributed residuals (not shown). Despite the seemingly high quality fit for m and $M_w^{agg}$ in Fig. 7A, the true value of ${\stackrel{‒}{\kappa }}_n$ increases dramatically as aggregation proceeds, although ${\stackrel{‒}{\kappa }}_w$ remains reasonably close to 1 throughout (data not shown). Thus, although the size-independent model fits the simulated {m, M_w} data well to within the precision of typical experimental data, the fitted value for τ_c is only a rough approximation to its true value.

Figure 7.

(A) Representative simulated aggregation kinetics (symbols) with size dependent condensation (Eq. 7-14, β_gn = 1000, β_cg = 20, x = 6, δ = 1); curves are fits to the size-independent model (Eq. 7-9, with ${\stackrel{‒}{\kappa }}_n={\stackrel{‒}{\kappa }}_w\equiv 1$). (B) comparison of fitted τ_g (gray), τ_n (white), and τ_c (black) values from the size-independent model versus the true values, based on results analogous to panel A but for a range of β_cg values. Asterisks indicate simulated data sets that were truncated at m = 0.95 before regression (see also details in text).

Fig. 7B further shows that for β_cg = ca. 10 or higher, deviations are found not only in τ_c, but in all three fitted parameters (τ_g,τ_n,τ_c). Thus, although the fits appeared to be good in all test cases, the fitted values of (τ_g,τ_n,τ_c) were inaccurate except when condensation was not dominant over chain polymerization (β_cg ~ 1 or smaller). The last two columns in Fig. 7B are for fits using a size-independent model of condensation, but with data truncated at low extents of reaction. In this case, accurate (τ_g,τ_n,τ_c) were obtained even when condensation is dominant (high β_cg). Intuitively, this is reasonable because at low extents of reaction the aggregate size distribution will lie relatively close to the nucleus size (x), and the assumption that all k_i,j values are the same as k_x,x is reasonable.

The above results clearly illustrate that aggregation kinetics monitored experimentally in terms of m and M_w can qualitatively identify whether condensation steps are appreciable, but that obtaining good fits to a kinetic model will not necessarily provide fitted parameter values that accurately reflect the true values for the system. Of course, true values of model parameters cannot be known a priori for an experimental system, and so it would not be possible to statistically distinguish these mechanisms in such a situation. As a result, it cannot be generally concluded that m and M_w kinetic data on their own will be sufficient to conclusively distinguish between alternative models for aggregate condensation. Preliminary results (not shown) indicate that this limitation might be overcome if one can experimentally measure higher moments of the distribution, as well as if one can accurately quantify sample polydispersity. In practice, this may remain an outstanding challenge because these quantities are difficult if not impossible to accurately quantify with currently available commercial equipment for the typical size ranges of soluble protein aggregates (~ 1 - 10² nm). Qualitatively, however, it may be possible to distinguish between different condensation mechanisms with information regarding aggregate morphology. For example, different types of condensation mechanisms may result in aggregates with different characteristic fractal structures.⁵¹ In such cases, this argues for the importance of using additional data, such as aggregate structure or morphology, when elucidating mechanistic details of aggregation.^16,51

3.3 LENP model applied to aggregation of aCgn

Figure 8 illustrates fits of the LENP model (Eq. 7-10) to experimental aggregation kinetics for α-chymotrypsinogen A (aCgn) monitored by size exclusion chromatography with inline static laser light scattering.⁵ The data are from two different solution conditions (summarized in the figure caption; additional details in ref. 5), and are plotted in the same format as Figures 2 and 3. In both cases the aggregates are soluble throughout the experimental time scale, and therefore n*→ ∞ for fitting with the LENP model. As was done in section 3.2, τ_n, τ_g, and τ_c were regressed for a range of integer values of δ and x to obtain the best least-squares fits to m(t) and $M_w^{agg}\left(t\right)$ data simultaneously. The best-fit values for each case, along with 95% confidence intervals are given in the caption to Figure 8.

Figure 8.

(adapted and reproduced with permission from ref. 5) Illustrative fits of the LENP model to two cases of experimental aggregation kinetics for aCgn. For both cases, the protein concentration (c₀) is 1 mg mL^-1 aCgn, and buffer conditions are pH 3.5, 10 mM sodium citrate buffer. The conditions differ in terms of incubation temperature and NaCl concentration: 60 °C with no NaCl (squares); 50 °C with 0.1 M NaCl (triangles). The curves are best-fits from least-squares regression vs. Eq. 7-10 with a size-independent condensation mechanism. The best-fit parameter values⁵ for the first data set (squares) are x = 3, δ = 1, τ_g = 0.1 ± 0.01 min, τ_n = 10³ ± 10² min, and τ_c > 10¹² min; the corresponding parameter values for the second data set (triangles) are x = 3, τ_g = 0.8 ± 0.1 min, τ_n = 500 ± 200 min, and τ_c = 0.1 ± 0.01 min. Panels A and B show the same data in two different formats, for easier comparion of the qualitative features from simulated data in Figures 2 and 3. The open symbols in panel A are $M_w^{agg}$ values from light scatering; the filled symbols are the corresponding m values from chromatography. Details of the experimental protocols are given elsewhere.⁵

Qualitative comparison with Fig. 2 and 3 shows that the selected conditions correspond to type II (squares) and type Ia (triangles) behavior. The qualitative features for the type Ia conditions cannot be produced without including condensation steps in the model (stage V, Fig. 1): for example, the pronounced upturn of $M_w^{agg}\mathrm{vs}.1-m$ in Figure 8B, and a concomitant, large increase in polydispersity⁵ (results not shown here). In quantitative terms, the best fit parameter values give β_gn ~ 10³ in both cases. They give β_cg ~ 10 and β_cg << 1, respectively, for the type Ia and II cases. These results are qualitatively and quantitatively consistent with the analysis and discussion in section 3.1. Finally, the different experimental conditions for aCgn in Figure 8 correspond to aggregates with qualitatively different morphology; the aggregates for the type II conditions in Figure 8 are linear polymers,^4,5 while those for the type Ia conditions are more globular and compact.⁵ These morphological differences are consistent with qualitative differences in growth mechanisms for limiting cases Ia and II in the LENP model. However, they do not provide sufficient information to discern additional details of the condensation mechanism (e.g., size dependent vs. size-independent k_i,j). A more global search of solution conditions that give rise to behaviors other than types II and Ia for aCgn is currently underway, and will be included as part of a future report.

4. Summary

This report presents an LENP model of nonnative protein aggregation that explicitly includes the contributions of aggregate-aggregate association or condensation. The model improves upon the previous LENP model²⁰ while maintaining its strengths and ability to capture a wide variety of experimental behaviors. The global behavior and application to simulated data are illustrated primarily using a size-independent condensation mechanism similar to that employed previously,^17,18 and to a lesser extent using a simple Smoluchowski, diffusion-limited condensation mechanism. Illustrative examples are also included via application of the LENP model to experimental aggregation kinetics of α-chymotrypsinogen A.⁵

The results illustrate a number of ways to qualitatively determine whether soluble aggregate growth occurs via chain polymerization, aggregate-aggregate condensation, or a combination of both. It is shown that this assessment is easily done by measuring both monomer loss (or mass percent conversion to aggregate) and weight-average molecular weight when monitoring aggregation kinetics. When high molecular weight aggregates remain soluble, moment-based kinetic equations provide a means to quantitatively separate the time scales or inverse rate coefficients for nucleation (τ_n), growth by chain polymerimation (τ_g), and condensation (τ_c). This requires time dependent data on aggregate molecular weight M_w, and cannot be done with only data for monomer concentration m. However, even regression against both m and M_w is not necessarily sufficient to distinguish between alternative models for condensation. Use of early time data to provide accurate values of τ_n and τ_g was also evaluated and found to provide reasonable estimates even when details of a condensation mechanism are unknown. The current LENP model is also easily adaptable to include more complex aggregate growth mechanisms.

5. Appendix

The dynamic material balances of monomer (m), nuclei (a_x) and larger aggregates (a_i,i>x) for the reaction scheme in Fig. 1 are given by Eq. A1-A3, assuming each step in Fig. 1 is an elementary reaction obeying mass-action kinetics, and stages 1 and 2 are pre-equilibrated. Rate coefficients and equilibrium constants are defined in Fig. 1 and are consistent with more detailed descriptions given in ref. 20.

$$\frac{dm}{dt}=-x{m}^x∕{\tau }_n-\delta m^{\delta }\sigma ∕{\tau }_g$$ (A1)

$$\frac{d{a}_x}{dt}=m^x∕{\tau }_n-a_x{m}^{\delta }∕{\tau }_g-k_{x,x}{C}_0{a_x}^2-\left(a_x\sum _{j=x}^{n\ast -1}{k}_{x,j}{C}_0{a}_j\right)$$ (A2)

$${\phantom{\mid }\frac{d{a}_i}{dt}\mid }_{\underset{i\in \left\{\mathit{even}\right\}}{x<i<n\ast }}=(a_{i-\delta }-a_i)m^{\delta }∕{\tau }_g-k_{i,i}{C}_0{a_i}^2-a_i\sum _{j=x}^{n\ast -1}{k}_{i,j}{C}_0{a}_j+\sum _{\underset{i-j\ge x}{j=x}}^{i∕2}{k}_{i-j,j}{C}_0{a}_{i-j}{a}_j$$ (A3)

Eqs. A1-3 are similar to expressions that were derived previously²⁰ except that terms are included to account for the consumption of nuclei by condensation steps, as well as formation and consumption of other aggregates through condensation. Symbols in the above equations are explained in Section 2.1, and are consistent with previous work.²⁰ The corresponding moment equations follow by taking weighted sums over da_i/dt from i = x to ∞, along with the model parameters defined in Sec. 2: Zeroth Moment:

$$\frac{d\sigma }{dt}=\sum _{i=x}^{\infty }\frac{d{a}_i}{dt}=\frac{m^x}{{\tau }_n}-\frac{1}{2{\tau }_c}(\sum _{i=x}^{\infty }{a}_i\sum _{j=x}^{\infty }{\kappa }_{i,j}{a}_j+\sum _{i=x}^{\infty }{\kappa }_{i,i}{a_i}^2)$$

$$\frac{d\sigma }{dt}\cong \frac{m^x}{{\tau }_n}-\frac{1}{2{\tau }_c}\left(\sum _{i=x}^{\infty }\sum _{j=x}^{\infty }{\kappa }_{i,j}{a}_i{a}_j\right)$$ (A4)

First Moment:

$$\frac{d{\lambda }_1}{dt}=\sum _{i=x}^{\infty }i\frac{d{a}_i}{dt}=\frac{d(1-m)}{dt}$$ (A5)

Second Moment:

$$\frac{d{\lambda }_2}{dt}=\sum _{i=x}^{\infty }{i}^2\frac{d{a}_i}{dt}$$

$$\frac{d{\lambda }_2}{dt}=\frac{x^2{m}^x}{{\tau }_n}+\frac{m^{\delta }}{{\tau }_g}\sum _{i=x}^{\infty }({(i+\delta )}^2-i^2)a_i+\frac{1}{{\tau }_c}(\sum _{i=x}^{\infty }i{a}_i\sum _{j=x}^{\infty }{\kappa }_{i,j}j{a}_j+\sum _{i=x}^{\infty }{\kappa }_{i,i}{\left(i{a}_i\right)}^2)$$

$$\frac{d{\lambda }_2}{dt}\cong x^2\frac{m^x}{{\tau }_n}+\frac{m^{\delta }({\delta }^2\sigma +2\delta {\lambda }_1)}{{\tau }_g}+\frac{1}{{\tau }_c}\left(\sum _{i=x}^{\infty }i{a}_i\sum _{j=x}^{\infty }{\kappa }_{i,j}j{a}_j\right)$$ (A6)

Eq. A4 and A6 are approximate only in that they neglect the terms ${\sum }_{i=x}^{\infty }{a}_i^2$ and ${\sum }_{i=x}^{\infty }{\left(i{a}_i\right)}^2$ respectively. These terms are due to the self association reaction a_i + a_i → a_i+i where two same-sized aggregates are consumed, and are negligible when the aggregate size distribution is not close to monodisperse, as is the case when nucleation is slow compared to growth via chain or condensation polymerization.