On one-dimensional nucleation and growth of "living" polymers I. Homogeneous nucleation

Advanced computational approaches to understand protein aggregation

Protein aggregation is a widespread phenomenon implicated in debilitating diseases like Alzheimer's, Parkinson's, and cataracts, presenting complex hurdles for the field of molecular biology. In this review, we explore the evolving realm of computational methods and bioinformatics tools that have revolutionized our comprehension of protein aggregation. Beginning with a discussion of the multifaceted challenges associated with understanding this process and emphasizing the critical need for precise predictive tools, we highlight how computational techniques have become indispensable for understanding protein aggregation. We focus on molecular simulations, notably molecular dynamics (MD) simulations, spanning from atomistic to coarse-grained levels, which have emerged as pivotal tools in unraveling the complex dynamics governing protein aggregation in diseases such as cataracts, Alzheimer's, and Parkinson's. MD simulations provide microscopic insights into protein interactions and the subtleties of aggregation pathways, with advanced techniques like replica exchange molecular dynamics, Metadynamics (MetaD), and umbrella sampling enhancing our understanding by probing intricate energy landscapes and transition states. We delve into specific applications of MD simulations, elucidating the chaperone mechanism underlying cataract formation using Markov state modeling and the intricate pathways and interactions driving the toxic aggregate formation in Alzheimer's and Parkinson's disease. Transitioning we highlight how computational techniques, including bioinformatics, sequence analysis, structural data, machine learning algorithms, and artificial intelligence have become indispensable for predicting protein aggregation propensity and locating aggregation-prone regions within protein sequences. Throughout our exploration, we underscore the symbiotic relationship between computational approaches and empirical data, which has paved the way for potential therapeutic strategies against protein aggregation-related diseases. In conclusion, this review offers a comprehensive overview of advanced computational methodologies and bioinformatics tools that have catalyzed breakthroughs in unraveling the molecular basis of protein aggregation, with significant implications for clinical interventions, standing at the intersection of computational biology and experimental research.

Sequence and structure-based peptides as potent amyloid inhibitors: A review

Misfolded and natively disordered globular proteins tend to aggregate together in an interwoven fashion to form fibrous, proteinaceous deposits referred to as amyloid fibrils. Formation and deposition of such insoluble fibrils are the characteristic features of a broad group of diseases, known as amyloidosis. Some of these proteins are known to cause several degenerative disorders in humans, such as Amyloid-Beta (Aβ) in Alzheimer's disease (AD), human Islet Amyloid Polypeptide (hIAPP, amylin) in type 2 diabetes, α-synuclein (α-syn) in Parkinson's disease (PD) and so on. The fact that these proteins do not share any significant sequence or structural homology in their native states make therapy quite challenging. However, it is observed that aggregation-prone proteins and peptides tend to adopt a similar type of secondary structure during the formation of fibrils. Rationally designed peptides can be a potent inhibitor that has been shown to disrupt the fibril structure by binding specifically to the amyloidogenic region(s) within a protein. The following review will analyze the inhibitory potency of both sequence-based and structure-based small peptides that have been shown to inhibit amyloidogenesis of proteins such as Aβ, human amylin, and α-synuclein.

Supramolecular polymerization: challenges and advantages of various methods in assessing the aggregation mechanism

Oligothiophenes with branched alkyl end groups show distinct aggregation in organic solvents. The process of supramolecular polymerization is assessed by three different methods (UV-vis absorption and fluorescence emission spectroscopy and dynamic light scattering) to exclude artifacts. An apparent dependence of the degree of aggregation on the concentration of the oligomers is observed. Above the upper limit of concentration (a lower micromolar range for the present class of compounds), experimental data delivered conflicting results and the concentration should not therefore be exceeded. Scanning force microscopy and molecular dynamics simulations confirm the formation of one-dimensional aggregates with presumably helical arrangement of the achiral monomers.

Physical principles of filamentous protein self-assembly kinetics

The polymerization of proteins and peptides into filamentous supramolecular structures is an elementary form of self-organization of key importance to the functioning biological systems, as in the case of actin biofilaments that compose the cellular cytoskeleton. Aberrant filamentous protein self-assembly, however, is associated with undesired effects and severe clinical disorders, such as Alzheimer's and Parkinson's diseases, which, at the molecular level, are associated with the formation of certain forms of filamentous protein aggregates known as amyloids. Moreover, due to their unique physicochemical properties, protein filaments are finding extensive applications as biomaterials for nanotechnology. With all these different factors at play, the field of filamentous protein self-assembly has experienced tremendous activity in recent years. A key question in this area has been to elucidate the microscopic mechanisms through which filamentous aggregates emerge from dispersed proteins with the goal of uncovering the underlying physical principles. With the latest developments in the mathematical modeling of protein aggregation kinetics as well as the improvement of the available experimental techniques it is now possible to tackle many of these complex systems and carry out detailed analyses of the underlying microscopic steps involved in protein filament formation. In this paper, we review some classical and modern kinetic theories of protein filament formation, highlighting their use as a general strategy for quantifying the molecular-level mechanisms and transition states involved in these processes.

Photoregulated Living Supramolecular Polymerization Established by Combining Energy Landscapes of Photoisomerization and Nucleation-Elongation Processes

The significant contribution of conventional living polymerization to polymer science assures that living supramolecular polymerization will also lead to a variety of novel phenomena and applications. However, the monomer scope still remains limited in terms of the self-assembly energy landscape; a kinetic trap that retards spontaneous nucleation has to be coupled with a supramolecular polymerization pathway, which is challenging to achieve by molecular design. Herein, we report a rational approach to addressing this issue. We combined the supramolecular polymerization and photoisomerization processes to build the energy landscape, wherein the monomer can be activated/deactivated by light irradiation. In this way, the supramolecular polymerization and kinetic trap can be independently designed in the energy landscape. When the "dormant" monomer was activated by light in the presence of the seed of the supramolecular polymer, the "activated" free monomer was polymerized at the termini of the seed in a chain-growth manner. As a result, we achieved supramolecular polymers with controlled lengths and a narrow polydispersity. Although photoisomerization has been extensively employed in supramolecular polymer chemistry, most studies have focused on the stimuli responsiveness. In this respect, the present study would provoke supramolecular chemists to revisit stimuli-responsive supramolecular polymer systems as potential candidates for devising living supramolecular polymerization.

Dynamics of protein aggregation and oligomer formation governed by secondary nucleation

The formation of aggregates in many protein systems can be significantly accelerated by secondary nucleation, a process where existing assemblies catalyse the nucleation of new species. In particular, secondary nucleation has emerged as a central process controlling the proliferation of many filamentous protein structures, including molecular species related to diseases such as sickle cell anemia and a range of neurodegenerative conditions. Increasing evidence suggests that the physical size of protein filaments plays a key role in determining their potential for deleterious interactions with living cells, with smaller aggregates of misfolded proteins, oligomers, being particularly toxic. It is thus crucial to progress towards an understanding of the factors that control the sizes of protein aggregates. However, the influence of secondary nucleation on the time evolution of aggregate size distributions has been challenging to quantify. This difficulty originates in large part from the fact that secondary nucleation couples the dynamics of species distant in size space. Here, we approach this problem by presenting an analytical treatment of the master equation describing the growth kinetics of linear protein structures proliferating through secondary nucleation and provide closed-form expressions for the temporal evolution of the resulting aggregate size distribution. We show how the availability of analytical solutions for the full filament distribution allows us to identify the key physical parameters that control the sizes of growing protein filaments. Furthermore, we use these results to probe the dynamics of the populations of small oligomeric species as they are formed through secondary nucleation and discuss the implications of our work for understanding the factors that promote or curtail the production of these species with a potentially high deleterious biological activity.

Estimation of the lag time in a subsequent monomer addition model for fibril elongation

The lag time for dock–lock fibril elongation can be estimated from kinetic parameters.

Asymptotic solutions of the Oosawa model for the length distribution of biofilaments

Nucleated polymerisation phenomena are general linear growth processes that underlie the formation of a range of biofilaments in nature, including actin and tubulin that are key components of the cellular cytoskeleton. The conventional theoretical framework for describing this process is the Oosawa model that takes into account homogeneous nucleation coupled to linear growth. In his original work, Oosawa provided an analytical solution to the total mass concentration of filaments; the time evolution of the full length distribution has, however, been challenging to access, in large part due to the nonlinear nature of the rate equations inherent in the description of such phenomena and to date analytical solutions for the filament distribution are known only in certain special cases. Here, by exploiting a technique based on the method of matched asymptotics, we present an analytical treatment of the Oosawa model that describes the shape of the length distribution of biofilaments reversibly growing through primary nucleation and filament elongation. Our work highlights the power of matched asymptotics for obtaining closed-form analytical solutions to nonlinear master equations in biophysics and allows us to identify the key time scales that characterize biological polymerization processes.

Modelling amyloid fibril formation kinetics: mechanisms of nucleation and growth

Amyloid and amyloid-like fibrils are self-assembling protein nanostructures, of interest for their robust material properties and inherent biological compatibility as well as their putative role in a number of debilitating mammalian disorders. Understanding fibril formation is essential to the development of strategies to control, manipulate or prevent fibril growth. As such, this area of research has attracted significant attention over the last half century. This review describes a number of different models that have been formulated to describe the kinetics of fibril assembly. We describe the macroscopic implications of mechanisms in which secondary processes such as secondary nucleation, fragmentation or branching dominate the assembly pathway, compared to mechanisms dominated by the influence of primary nucleation. We further describe how experimental data can be analysed with respect to the predictions of kinetic models.

From macroscopic measurements to microscopic mechanisms of protein aggregation

The ability to relate bulk experimental measurements of amyloid formation to the microscopic assembly processes that underlie protein aggregation is critical in order to achieve a quantitative understanding of this complex phenomenon. In this review, we focus on the insights from classical and modern theories of linear growth phenomena and discuss how theory allows the roles of growth and nucleation processes to be defined through the analysis of experimental in vitro time courses of amyloid formation. Moreover, we discuss the specific signatures in the time course of the reactions that correspond to the actions of primary and secondary nucleation processes, and outline strategies for identifying and characterising the nature of the dominant process responsible in each case for the generation of new aggregates. We highlight the power of a global analysis of experimental time courses acquired under different conditions, and discuss how such an analysis allows a rigorous connection to be established between the macroscopic measurements and the rates of the individual microscopic processes that underlie the phenomenon of amyloid formation.

Nucleated polymerization with secondary pathways. I. Time evolution of the principal moments

Self-assembly processes resulting in linear structures are often observed in molecular biology, and include the formation of functional filaments such as actin and tubulin, as well as generally dysfunctional ones such as amyloid aggregates. Although the basic kinetic equations describing these phenomena are well-established, it has proved to be challenging, due to their non-linear nature, to derive solutions to these equations except for special cases. The availability of general analytical solutions provides a route for determining the rates of molecular level processes from the analysis of macroscopic experimental measurements of the growth kinetics, in addition to the phenomenological parameters, such as lag times and maximal growth rates that are already obtainable from standard fitting procedures. We describe here an analytical approach based on fixed-point analysis, which provides self-consistent solutions for the growth of filamentous structures that can, in addition to elongation, undergo internal fracturing and monomer-dependent nucleation as mechanisms for generating new free ends acting as growth sites. Our results generalise the analytical expression for sigmoidal growth kinetics from the Oosawa theory for nucleated polymerisation to the case of fragmenting filaments. We determine the corresponding growth laws in closed form and derive from first principles a number of relationships which have been empirically established for the kinetics of the self-assembly of amyloid fibrils.

Nucleated polymerisation in the presence of pre-formed seed filaments

We revisit the classical problem of nucleated polymerisation and derive a range of exact results describing polymerisation in systems intermediate between the well-known limiting cases of a reaction starting from purely soluble material and for a reaction where no new growth nuclei are formed.

Fitting neurological protein aggregation kinetic data via a 2-step, minimal/"Ockham's razor" model: the Finke-Watzky mechanism of nucleation followed by autocatalytic surface growth

The aggregation of proteins has been hypothesized to be an underlying cause of many neurological disorders including Alzheimer's, Parkinson's, and Huntington's diseases; protein aggregation is also important to normal life function in cases such as G to F-actin, glutamate dehydrogenase, and tubulin and flagella formation. For this reason, the underlying mechanism of protein aggregation, and accompanying kinetic models for protein nucleation and growth (growth also being called elongation, polymerization, or fibrillation in the literature), have been investigated for more than 50 years. As a way to concisely present the key prior literature in the protein aggregation area, Table 1 in the main text summarizes 23 papers by 10 groups of authors that provide 5 basic classes of mechanisms for protein aggregation over the period from 1959 to 2007. However, and despite this major prior effort, still lacking are both (i) anything approaching a consensus mechanism (or mechanisms), and (ii) a generally useful, and thus widely used, simplest/"Ockham's razor" kinetic model and associated equations that can be routinely employed to analyze a broader range of protein aggregation kinetic data. Herein we demonstrate that the 1997 Finke-Watzky (F-W) 2-step mechanism of slow continuous nucleation, A --> B (rate constant k1), followed by typically fast, autocatalytic surface growth, A + B --> 2B (rate constant k2), is able to quantitatively account for the kinetic curves from all 14 representative data sets of neurological protein aggregation found by a literature search (the prion literature was largely excluded for the purposes of this study in order provide some limit to the resultant literature that was covered). The F-W model is able to deconvolute the desired nucleation, k1, and growth, k2, rate constants from those 14 data sets obtained by four different physical methods, for three different proteins, and in nine different labs. The fits are generally good, and in many cases excellent, with R2 values >or=0.98 in all cases. As such, this contribution is the current record of the widest set of protein aggregation data best fit by what is also the simplest model offered to date. Also provided is the mathematical connection between the 1997 F-W 2-step mechanism and the 2000 3-step mechanism proposed by Saitô and co-workers. In particular, the kinetic equation for Saitô's 3-step mechanism is shown to be mathematically identical to the earlier, 1997 2-step F-W mechanism under the 3 simplifying assumptions Saitô and co-workers used to derive their kinetic equation. A list of the 3 main caveats/limitations of the F-W kinetic model is provided, followed by the main conclusions from this study as well as some needed future experiments.

Mechanisms of protein fibril formation: nucleated polymerization with competing off-pathway aggregation

The formation of protein fibrils, and in particular amyloid fibrils, underlies many human diseases. Understanding fibril formation mechanisms is important for understanding disease pathology, but fibril formation kinetics can be complicated, making the relationship between experimental observables and specific mechanisms unclear. Here we examine one often-proposed fibril formation mechanism, nucleated polymerization with off-pathway aggregation. We use the characteristics of this mechanism to derive three tests that can be performed on experimental data to identify it. We also find that this mechanism has an especially striking feature: although increasing protein concentrations generally cause simple nucleated polymerizations to reach completion faster, they cause nucleated polymerizations with off-pathway aggregation to reach completion more slowly when the protein concentration becomes too high.

The kinetics of nucleated polymerizations at high concentrations: amyloid fibril formation near and above the "supercritical concentration"

The formation of amyloid and other types of protein fibrils is thought to proceed by a nucleated polymerization mechanism. One of the most important features commonly associated with nucleated polymerizations is a strong dependence of the rate on the concentration. However, the dependence of fibril formation rates on concentration can weaken and nearly disappear as the concentration increases. Using numerical solutions to the rate equations for nucleated polymerization and analytical solutions to some limiting cases, we examine this phenomenon and show that it is caused by the concentration approaching and then exceeding the equilibrium constant for dissociation of monomers from species smaller than the nucleus, a quantity we have named the "supercritical concentration". When the concentration exceeds the supercritical concentration, the monomer, not the nucleus, is the highest-energy species on the fibril formation pathway, and the fibril formation reaction behaves initially like an irreversible polymerization. We also derive a relation that can be used in a straightforward method for determining the nucleus size and the supercritical concentration from experimental measurements of fibril formation rates.

Analysis of protein aggregation kinetics

Given a set of kinetic data, then, the preceding discussions suggest the following approach to its analysis. 1. For purposes of establishing the reaction, ignore the final stages and concentrate on the initial 10-20% of the reaction at first. A globally optimized model may be based on a faulty assumption for the initial steps. Thus, although the whole data set may look reasonably well fit, the reaction could be misrepresented, and thus the fit unhelpful if accuracy at the later stages has come at the expense of the initial phase of the reaction. 2. What is the time course of the initial reaction? (A) Is the reaction exponential? Exponential growth gives dramatic lag times (see Fig. 3), whereas nonexponential "lag times" have a visible signal from time 0 (i.e., Fig. 2). If the data set shows the abrupt appearance of signals after a period of quiescence, the chances are excellent that the time course is exponential. High sensitivity measurement of the signal at times during the lag phase should be used to confirm the exponential nature quantitatively. Exponential reactions mean a secondary pathway is operative. (a) A cascade (tn) can look similar to an exponential, but may proceed from a multistep single-path reaction. Thus the exponential needs to be ascertained with some accuracy. (b) It is possible that some or all of the lag results from a stochastic process, i.e., formation of a single nucleus being observed. This, however, is likely to be accompanied by a secondary process, as few techniques are sensitive enough to detect a single polymer at a time, and having one nucleus form many polymers is a hallmark of a secondary process. Thus, the reproducibility of the kinetics must be established to rule out stochastics. If data show wide variation, stochastic methods as described earlier may be employed. (c) Given a secondary process, one must separate the primary nucleation process from the secondary process (by stochastic means or by use of the product B2A, as described earlier). (B) If the reaction does not begin with an exponential, is it parabolic? If so, it falls in the general class of linear polymerizations. 3. What is the concentration dependence of the reaction(s)? This will separate nucleation processes from growth, and so on. 4. If the initial reaction is neither exponential nor parabolic, a reaction mechanism needs to be proposed and evaluated. Solving the resulting equations is best done by linearization, which has the best chance of giving equations whose solutions and their sensitivity to parameters are readily understood. If this proves fruitful, full numeric solutions may be useful. 5. At this point, the full reaction may be considered to completion. 6. The physical basis of the description (sizes of parameters and their dependencies) needs to be finally considered to ensure that the mathematical success of the description rests on tenable physical grounds.

Theoretical description of the spatial dependence of sickle hemoglobin polymerization

We have generalized the double nucleation mechanism of Ferrone et al. (Ferrone, F. A., J. Hofrichter, H. Sunshine, and W. A. Eaton. 1980. Biophys. J. 32:361-377; Ferrone, F. A., J. Hofrichter, and W. A. Eaton. 1985. J. Mol. Biol. 183:611-631) to describe the spatial dependence of the radial growth of polymer domains of sickle hemoglobin. Although this extended model requires the consideration of effects such as monomer diffusion, which are irrelevant to a spatially uniform description, no new adjustable parameters are required because diffusion constants are known independently. We find that monomer diffusion into the growing domain can keep the net unpolymerized monomer concentration approximately constant, and in that limit we present an analytic solution of the model. The model shows the features reported by Basak, S., F. A. Ferrone, and J. T. Wang (1988. Biophys J. 54:829-843) and provides a new means of determining the rate of polymer growth. When spatially integrated, the model exhibits the exponential growth seen in previous studies, although molecular parameters derived from analysis of the kinetics assuming uniformity must be modified in some cases to account for the spatially nonuniform growth. The model developed here can be easily adapted to any spatially dependent polymerization process.

Kinetics of sickle hemoglobin polymerization. III. Nucleation rates determined from stochastic fluctuations in polymerization progress curves

The polymerization kinetics of sickle cell hemoglobin are found to exhibit stochastic variations when observed in very small volumes (approximately 10(-10) cm3). The distribution of progress curves has been measured at several temperatures for a 4.50 mM-hemoglobin S sample using a laser-photolysis, light-scattering technique. The progress curves at a given temperature are superimposable when translated along the time axis, showing that the variability of the kinetic progress curves results primarily from fluctuations in the time at which polymerization is initiated. The shapes of the initial part of the progress curves are well-fitted using the functional form I(t) = Io + As exp (Bt), derived from a dual nucleation model. When the distribution of the measured tenth times is broad, the rate of homogeneous nucleation can be obtained by fitting the exponential tail of the distribution. As the distribution sharpen, the rate of homogeneous nucleation can be estimated by modelling the width of the distribution function using a simple Monte-Carlo simulation of the polymerization kinetics. Using the rates of homogeneous nucleation obtained from the distributions, the rates of heterogeneous nucleation and polymer growth can be obtained from the experimental parameters As and B. The resulting nucleation rates are roughly 1000 times greater than those obtained from an analysis of bulk kinetic data. The results provide strong support for the dual-nucleation mechanism and show that the distribution of progress curves provides a powerful independent method for measuring the rate of homogeneous nucleation and thereby obtaining values for the other principal rates of the mechanism.

On one-dimensional nucleation and growth of "living" polymers. II. Growth at constant monomer concentration

An analytical solution is given for the kinetics of reversible homogeneous one-dimensional growth, assuming that all association rate constants have the same value k, that all dissociation rate constants are likewise equal to k, and that the monomer concentration has a constant value, C. Such growth tends to generate a maximally polydisperse ("white") distribution of cluster concentrations ci, all approaching a limiting value equal to that of the critical nucleus, cn. Continued growth merely increases the range of cluster sizes over which this white distribution applies. A simple expression is obtained for the flux sigma infinity i = n dci/dt, which becomes constant and equal to (kC - k)cn. The monomer uptake increases with time, and is given approximately by (kC - k)2cnt.