Home | About | Journals | Submit | Contact Us | Français |

**|**HHS Author Manuscripts**|**PMC2683240

Formats

Article sections

- Abstract
- 1. Introduction
- 2. Model Description & Derivations
- 3. Results & Discussion
- 4. Summary
- References

Authors

Related links

J Phys Chem B. Author manuscript; available in PMC 2010 May 14.

Published in final edited form as:

PMCID: PMC2683240

NIHMSID: NIHMS111057

Department of Chemical Engineering, 150 Academy St., Colburn Laboratory, University of Delaware, Newark, DE 19716

The publisher's final edited version of this article is available at J Phys Chem B

See other articles in PMC that cite the published article.

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

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 - 10^{3} 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 report^{20} 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 al^{17} and Speed et al^{18} 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* knowledge^{16} 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}

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

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 **...**

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 (

The following derivations are based on the reaction scheme in Fig. 1, and employ the same nomenclature as previous work^{20} to the extent possible here. Characteristic time scales are defined for nucleation $({\tau}_{n}\equiv {\tau}_{n}^{\left(0\right)}{f}_{R}^{-x}{({C}_{ref}\u2215{C}_{0})}^{x-1})$, growth via monomer addition $({\tau}_{g}\equiv {\tau}_{g}^{\left(0\right)}{f}_{R}^{-\delta}{({C}_{ref}\u2215{C}_{0})}^{\delta})$, and condensation $({\tau}_{c}\equiv \frac{{\tau}_{c}^{\left(0\right)}{C}_{ref}}{{C}_{0}})$ (see also Appendix). In these definitions, *f _{R}* = [

The above definitions along with the derivations elsewhere^{20} 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,j}C*

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

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

(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)

$${\phantom{\mid}\frac{d{a}_{i}}{d\theta}\mid}_{\underset{i\in \left\{\mathit{even}\right\}}{x<i<n\ast}}={\beta}_{gn}({a}_{i-1}-{a}_{i}){m}^{\delta}-{\beta}_{cg}{\beta}_{gn}{\kappa}_{i,i}{{a}_{i}}^{2}-{\beta}_{cg}{\beta}_{gn}{a}_{i}\sum _{j=x}^{n\ast -1}{\kappa}_{i,j}{a}_{j}+{\beta}_{cg}{\beta}_{gn}\sum _{j=x}^{i\u22152}{\kappa}_{i-j,j}{a}_{i-j}{a}_{j}$$

(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*])/*C*_{0}, with contributions from [*R _{i}*] neglected for

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.

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

Eq. 4 also relates ${M}_{w}^{agg}$ and ${M}_{n}^{agg}$ to the first and second moments of the soluble aggregate size distribution (λ_{1} and λ_{2}, 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** → ∞, λ_{1} 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 previously^{20} (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)

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)

$$\frac{d\sigma}{dt}=\frac{{m}^{x}}{{\tau}_{n}}-\frac{{\stackrel{\u2012}{\kappa}}_{n}}{2{\tau}_{c}}{\sigma}^{2}$$

(8)

$$\frac{d{\lambda}_{2}}{dt}={x}^{2}\frac{{m}^{x}}{{\tau}_{n}}+\frac{1}{{\tau}_{g}}{m}^{\delta}({\delta}^{2}\sigma +2\delta (1-m))+\frac{{\stackrel{\u2012}{\kappa}}_{w}}{{\tau}_{c}}{(1-m)}^{2}$$

(9)

with

$${\stackrel{\u2012}{\kappa}}_{n}\equiv \frac{\sum _{i=x}^{\infty}\sum _{j=x}^{\infty}{\kappa}_{i,j}{a}_{i}{a}_{j}}{\sum _{i=x}^{\infty}\sum _{j=x}^{\infty}{a}_{i}{a}_{j}}$$

(10a)

$${\stackrel{\u2012}{\kappa}}_{w}\equiv \frac{\sum _{i=x}^{\infty}\sum _{j=x}^{\infty}{\kappa}_{i,j}\left(i{a}_{i}\right)\left(j{a}_{j}\right)}{\sum _{i=x}^{\infty}\sum _{j=x}^{\infty}\left(i{a}_{i}\right)\left(j{a}_{j}\right)}$$

(10b)

and with the first moment (λ_{1} replaced by *m*. Eq. 10 defines number-average and weight-average κ* _{i,j}* values (${\stackrel{\u2012}{\kappa}}_{n}$ and ${\stackrel{\u2012}{\kappa}}_{w}$, respectively). In the most general case, ${\stackrel{\u2012}{\kappa}}_{n}$ and ${\stackrel{\u2012}{\kappa}}_{w}$ are not constant because they depend on the aggregate distribution {

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 elsewhere^{20} must instead be used.

As a test case to explore the effects of a physically plausible size dependence for κ* _{i,j}*, a difffusion-limited Smoluchowski model

$${k}_{i,j}=4\pi {N}_{A}({D}_{i}+{D}_{j})({R}_{i}+{R}_{j})f$$

(11)

*N _{A}* is Avogadro's number;

$${k}_{i,j}=\frac{2{N}_{A}{k}_{B}T}{3\eta}(\frac{1}{{R}_{i}}+\frac{1}{{R}_{j}})({R}_{i}+{R}_{j})$$

(12)

where *k _{B}* is Boltzmann's constant,

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

(13)

${\stackrel{\u2012}{\kappa}}_{n}$ and ${\stackrel{\u2012}{\kappa}}_{w}$ 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

$$\mu =\frac{\sum _{i=x}^{\infty}i{a}_{i}}{\sum _{i=x}^{\infty}{a}_{i}}=\frac{{\lambda}_{1}}{\sigma}=\frac{{M}_{n}^{\mathit{agg}}}{{M}_{\mathit{mon}}}$$

(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, λ_{1} (or *m*) and λ_{2}.

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

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

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 *d*σ/*dt*; (3) weight-average molecular weight of soluble aggregates, ${M}_{w}^{agg}\u2215{M}_{mon}$; (4) aggregate polydispersity, ${M}_{w}^{agg}\u2215{M}_{n}^{agg}$. 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).

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 and33 illustrate the qualitative behaviors in terms of *m*(*t*), σ(*t*), ${M}_{w}^{agg}\u2215{M}_{mon}\phantom{\rule{thickmathspace}{0ex}}\mathrm{vs}.\phantom{\rule{thinmathspace}{0ex}}(1-m)$, and ${M}_{w}^{agg}\u2215{M}_{n}^{agg}\phantom{\rule{thickmathspace}{0ex}}\mathrm{vs}.\phantom{\rule{thinmathspace}{0ex}}(1-m)$. 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 and3,3, *t* is scaled by *t*_{50} in order to more easily compare profiles with greatly different absolute time scales; *t*_{50} is defined by *m*(*t* = *t*_{50}) = 0.5.

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 previously^{20} 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 ${M}_{w}^{agg}\u2215{M}_{mon}$ 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 and33 correspond to a state point in 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, ${\stackrel{\u2012}{\kappa}}_{n}={\stackrel{\u2012}{\kappa}}_{w}\equiv 1$ for all points). Panel A: varying β_{gn} and **...**

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),3A), *k*_{obs} scaling with *C*_{0} to at least the first power, ${M}_{w}^{agg}\u2215{M}_{mon}$ increasing as (1-*m*) raised to a power much greater than 1 (Figs 2B, ,3B),3B), and high polydispersity values (Figs. 2D, ,3D).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).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 (

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)3A) and *k*_{obs} ~ *C*_{0}^{x-1}; and essentially undetectably low soluble aggregate concentrations (Fig. 2C). Little or no information regarding ${M}_{w}^{agg}\u2215{M}_{mon}$ 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)3A) and *k*_{obs} ~ *C*_{0}^{x-1}; low values of ${M}_{w}^{agg}\u2215{M}_{mon}$, and ${M}_{w}^{agg}\u2215{M}_{mon}$ (Figs. 2D, ,3D).3D). σ increases monotonically to a relatively large plateau value (Figs. 2C, ,3C)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)3A) and *k*_{obs} ~ *C*_{0}^{(x+†-1)/2}; ${M}_{w}^{agg}\u2215{M}_{mon}$ scales linearly with (1-*m*) once *m* is significantly less than 1 (Figs. 2B, ,3B,3B, and discussion below); low polydispersity that depends only weakly on extent of reaction (Figs. 2D, ,3D)3D) σ increases monotonically to a plateau value (Figs. 2C, ,3C)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 λ_{1} = (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 (${M}_{w}^{agg}\u2215{M}_{n}^{agg}$) 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 - -4.4. 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 tailspike^{18} 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).

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×*t*_{50}) 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 (*R*^{2}) 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 *R*^{2} 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.

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 (*C*_{0}).^{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 *C*_{0} 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, *R*^{2} 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 ${M}_{w}^{agg}\u2215{M}_{n}^{agg}=1.1$ (*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, σ^{2}) 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 $({M}_{w}^{agg}\u2215{M}_{n}^{agg}~1.1-1.2)$.^{4}^{,}^{5} The results in Fig. 2C suggest an additional criterion for neglecting condensation is that *M _{w}^{agg}* scales linearly with (1-

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 *R*^{2} > 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 ${\stackrel{\u2012}{\kappa}}_{n}$ increases dramatically as aggregation proceeds, although ${\stackrel{\u2012}{\kappa}}_{w}$ 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.

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 - 10^{2} 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}

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 and3.3. 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.

(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 (*c*_{0}) is 1 mg mL^{-1} aCgn, and buffer conditions are pH 3.5, 10 **...**

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 ${M}_{w}^{agg}\phantom{\rule{thinmathspace}{0ex}}\mathrm{vs}.\phantom{\rule{thinmathspace}{0ex}}1-m$ in Figure 8B, and a concomitant, large increase in polydispersity^{5} (results not shown here). In quantitative terms, the best fit parameter values give β_{gn} ~ 10^{3} 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

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 model^{20} 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.

Financial support from Merck & Co. (YL) and the National Institutes of Health (CJR; grant no. R01 EB006006) is gratefully acknowledged.

The dynamic material balances of monomer (*m*), nuclei (*a _{x}*) and larger aggregates (

$$\frac{dm}{dt}=-x{m}^{x}\u2215{\tau}_{n}-\delta {m}^{\delta}\sigma \u2215{\tau}_{g}$$

(A1)

$$\frac{d{a}_{x}}{dt}={m}^{x}\u2215{\tau}_{n}-{a}_{x}{m}^{\delta}\u2215{\tau}_{g}-{k}_{x,x}{C}_{0}{{a}_{x}}^{2}-\left({a}_{x}\sum _{j=x}^{n\ast -1}{k}_{x,j}{C}_{0}{a}_{j}\right)$$

(A2)

$${\phantom{\mid}\frac{d{a}_{i}}{dt}\mid}_{\underset{i\in \left\{\mathit{even}\right\}}{x<i<n\ast}}=({a}_{i-\delta}-{a}_{i}){m}^{\delta}\u2215{\tau}_{g}-{k}_{i,i}{C}_{0}{{a}_{i}}^{2}-{a}_{i}\sum _{j=x}^{n\ast -1}{k}_{i,j}{C}_{0}{a}_{j}+\sum _{\underset{i-j\ge x}{j=x}}^{i\u22152}{k}_{i-j,j}{C}_{0}{a}_{i-j}{a}_{j}$$

(A3)

Eqs. A1-3 are similar to expressions that were derived previously^{20} 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}*/

$$\frac{d\sigma}{dt}=\sum _{i=x}^{\infty}\frac{d{a}_{i}}{dt}=\frac{{m}^{x}}{{\tau}_{n}}-\frac{1}{2{\tau}_{c}}(\sum _{i=x}^{\infty}{a}_{i}\sum _{j=x}^{\infty}{\kappa}_{i,j}{a}_{j}+\sum _{i=x}^{\infty}{\kappa}_{i,i}{{a}_{i}}^{2})$$

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

(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}* +

(1) Fink AL. Folding & Design. 1998;3:R9. [PubMed]

(2) Roberts CJ. Biotechnology and Bioengineering. 2007;98:927. [PubMed]

(3) Chi EY, Krishnan S, Randolph TW, Carpenter JF. Pharmaceutical Research. 2003;20:1325. [PubMed]

(4) Weiss WF, IV, Hodgdon TK, Kaler EW, Lenhoff AM, Roberts CJ. Biophys J. 2007;93:4392. [PubMed]

(5) Li Y, Weiss WF, IV, Roberts CJ. J Pharm Sci. Submitted.

(6) Weiss WF, IV, Young TM, Roberts CJ. J Pharm Sci. 2008

(7) Cromwell MEM, Hilario E, Jacobson F. AAPS Journal. 2006;8:E572. [PMC free article] [PubMed]

(8) Wang W. International Journal of Pharmaceutics. 2005;289:1. [PubMed]

(9) Dobson CM. Seminars in Cell & Developmental Biology. 2004;15:3. [PubMed]

(10) Uversky VN, Fink AL. Biochimica et Biophysica Acta, Proteins and Proteomics. 2004;1698:131. [PubMed]

(11) Rosenberg AS. AAPS Journal. 2006;8:E501. [PMC free article] [PubMed]

(12) Purohit VS, Middaugh CR, Balasubramanian SV. J Pharm Sci. 2006;95:358. [PMC free article] [PubMed]

(13) Philo JS. Aaps J. 2006;8:E564. [PMC free article] [PubMed]

(14) Goetz H, Kuschel M, Wulff T, Sauber C, Miller C, Fisher S, Woodward C. J Biochem Biophys Methods. 2004;60:281. [PubMed]

(15) Andrews JM, Roberts CJ. Biochemistry. 2007;46:7558. [PubMed]

(16) Pallitto MM, Murphy RM. Biophys J. 2001;81:1805. [PubMed]

(17) Modler AJ, Gast K, Lutsch G, Damaschun G. J Mol Biol. 2003;325:135. [PubMed]

(18) Speed MA, King J, Wang DIC. Biotechnology And Bioengineering. 1997;54:333. [PubMed]

(19) Andrews JM, Weiss WF, IV, Roberts CJ. Biochemistry. 2008;47:2397. [PubMed]

(20) Andrews JM, Roberts CJ. J Phys Chem B. 2007;111:7897. [PubMed]

(21) Roberts CJ. Journal of Physical Chemistry B. 2003;107:1194.

(22) Roberts CJ. Non Native Protein Aggregation: Pathways, Kinetics, and Shelf Life Prediction. In: Murphy RM, Tsai AM, editors. Misbehaving Proteins: Protein Misfolding, Aggregation, and Stability. Springer; New York: 2006. p. 17.

(23) Ferrone F. Methods in Enzymology. 1999;309:256. [PubMed]

(24) Oosawa F, Asakura S. Thermodynamics of the Polymerization of Proteins. Academic Press; London: 1975.

(25) Mahler HC, Friess W, Grauschopf U, Kiese S. J Pharm Sci. 2008 [PubMed]

(26) Ramkrishna D. Population Balances: Theory and Applications to Particulate Systems in Engineering. 1st edition Academic Press; New York: 2007.

(27) Lee CC, Nayak A, Sethuraman A, Belfort G, McRae GJ. Biophys J. 2007;92:3448. [PubMed]

(28) Chen SM, Ferrone FA, Wetzel R. Proceedings Of The National Academy Of Sciences Of The United States Of America. 2002;99:11884. [PubMed]

(29) Powers ET, Powers DL. Biophys J. 2006;91:122. [PubMed]

(30) Chi EY, Krishnan S, Kendrick BS, Chang BS, Carpenter JF, Randolph TW. Protein Science. 2003;12:903. [PubMed]

(31) Gibson TJ, Murphy RM. Biochemistry. 2005;44:8898. [PubMed]

(32) Kim JR, Gibson TJ, Murphy RM. Biotechnol Prog. 2006;22:605. [PubMed]

(33) Chi EY, Kendrick BS, Carpenter JF, Randolph TW. J Pharm Sci. 2005;94:2735. [PubMed]

(34) Kurganov BI. Biochemistry (Mosc) 1998;63:364. [PubMed]

(35) Liu J, Andya JD, Shire SJ. AAPS Journal. 2006;8:E580. [PMC free article] [PubMed]

(36) Bourhim M, Kruzel M, Srikrishnan T, Nicotera T. J Neurosci Methods. 2007;160:264. [PubMed]

(37) LeVine H., 3rd Protein Sci. 1993;2:404. [PubMed]

(38) Kendrick BS, Cleland JL, Lam X, Nguyen T, Randolph TW, Manning MC, Carpenter JF. J Pharm Sci. 1998;87:1069. [PubMed]

(39) Webb JN, Webb SD, Cleland JL, Carpenter JF, Randolph TW. Proc Natl Acad Sci U S A. 2001;98:7259. [PubMed]

(40) Hiemenz PC. Polymer Chemistry: The Basic Concepts. Marcel Dekker; New York: 1984.

(41) Wen J, Arakawa T, Philo JS. Anal Biochem. 1996;240:155. [PubMed]

(42) Roberts CJ, Darrington RT, Whitley MB. J Pharm Sci. 2003;92:1095. [PubMed]

(43) Barzykin AV, Shushin AI. Biophysical Journal. 2001;80:2062. [PubMed]

(44) Smoluchowski M. v. Z. Phys. Chem. 1917;92:129.

(45) Sandkühler P. AIChE Journal. 2003;49:1542.

(46) Hilbe JM. Negative Binomial Regression. Cambridge University Press; Cambridge, UK: 2007.

(47) Walpole RE. Probability & Statistics for Engineers & Scientists. 8th ed. Pearson, Prentice Hall; Upper saddle River, NJ: 2006.

(48) Tsai AM, van Zanten JH, Betenbaugh MJ. Biotechnol Bioeng. 1998;59:273. [PubMed]

(49) Ignatova Z, Gierasch LM. Biochemistry. 2005;44:7266. [PubMed]

(50) Buswell AM, Middelberg APJ. Biotechnology And Bioengineering. 2003;83:567. [PubMed]

(51) Meakin P. Annual Review of Physical Chemistry. 1988;39:237.

PubMed Central Canada is a service of the Canadian Institutes of Health Research (CIHR) working in partnership with the National Research Council's national science library in cooperation with the National Center for Biotechnology Information at the U.S. National Library of Medicine(NCBI/NLM). It includes content provided to the PubMed Central International archive by participating publishers. |