Stochastic models for cell kinetics

A survey is given of branching process type methods in cell kinetics. Some results are given that allow circadian rhythm and do not require complete independence between cells. Some more classical results on balanced exponential growth are given and some comments are made on flow microfluorometry.     

Simple method of recording the kinetics of cell agglutination and aggregation

The light scattering measure method for studying the kinetics of cell agglutination and aggregation under conditions of continuous and uniform stirring of suspension has been suggested.     

Transitivity,pattern-reversal,engulfment and duality in exchange-type cell aggregation kinetics

Simulations of cell-sorting phenomena in embryogenesis describable by means of the differential adhesion mechanism are further studied using the exchange model. In this study, more than two cell-types are employed and results nicely complement the successes of earlier work with only two cell-types. Again, it is noticed that “central clumping” does not usually appear in final distributions. In addition, the phenomena of transitivity, pattern-reversal and the engulfment of one tissue by another are studied. The first is shown to be a trivial consequence of Steinberg's stochastic version of differential adhesion. Also, it is shown that the exchange model successfully simulates pattern-reversal by the reversal of the relative adhesion values due to the time-in-culture reversal of the effect of adhesivity of trypsin-assisted dissociation of tissues. The connection between the occurrence of pattern reversal and the absolute and relative positions of tissue types in the adhesion hierarchy is explained in terms on the exchange model. It is found that the model, as it stands, does not seem to give a satisfactory basis for the simulation of engulfment. Suggestions for the extension of the exchange principle are made which will perhaps rectify this limitation. It is felt that any real system which exhibits engulfment will also exhibit central clumping, and conversely. The importance of the figure-ground ambiguity we have called “duality” is again stressed. If this ambiguity is not taken into account in experimental work, interpretation of many results on cell-sorting are rendered meaningless.     

Interpretation of concentration-dependence in aggregation kinetics

Aggregation processes are analyzed by two kinetic models, the random polymerization model and the nucleation-dependent polymerization model. A kinetic equation for the random polymerization model can be derived analytically, revealing the relation between the initial monomer concentration ([M]0), the rate constant (k(a)), time (t), the yield of detectable aggregate ([F]), and the critical aggregation number (m). However, time-course curves for the nucleation-dependent polymerization model can be obtained by numerical calculation. It is found that lag time (t(d)) and half-time (t1/2) are proportional to [M](-1) in the random polymerization model, while t(d) and t1/2 are proportional to [M1](-s) (1 < s < n; n is nucleus size) at the lower concentration and are less dependent on [M1] at the higher concentration in the nucleation-dependent polymerization model.     

Surfactant effects on amyloid aggregation kinetics

There is strong experimental evidence of the influence of surfactants (e.g., fatty acids) on the kinetics of amyloid fibril formation. However, the structures of mixed assemblies and interactions between surfactants and fibril-forming peptides are still not clear. Here, coarse-grained simulations are employed to study the aggregation kinetics of amyloidogenic peptides in the presence of amphiphilic lipids. The simulations show that the lower the fibril formation propensity of the peptides, the higher the influence of the surfactants on the peptide self-assembly kinetics. In particular, the lag phase of weakly aggregating peptides increases because of the formation of mixed oligomers, which are promoted by hydrophobic interactions and favorable entropy of mixing. A transient peak in the number of surfactants attached to the growing fibril is observed before reaching the mature fibril in some of the simulations. This peak originates from transient fibrillar defects consisting of exposed hydrophobic patches on the fibril surface, which provide a possible explanation for the temporary maximum of fluorescence observed sometimes in kinetic traces of the binding of small-molecule dyes to amyloid fibrils.     

The kinetics of aggregation phenomena. I. Minimal models for patch formation of lymphocyte membranes.

Numerous biological processes involve the assembly of one or more monomers into aggregates or networks of interconnected units. In this paper we present the initial aspects of a mathematical theory for network formation on lymphocyte membranes. We assume the fluid mosaic membrane model is valid, that a lymphocyte possesses a homogeneous set of mobile but membrane bound receptors and that these receptors can form bimolecular complexes with antigen. We show that these complexes tend to aggregate and derive expressions for their size distribution as a function of time, antigen valence and concentration, and antigenreceptor affinity.At early times, the mass of the system (receptors plus antigen) is in very small aggregates. However under appropriate conditions, a critical time is reached at which they coalesce in such a way that the mass shifts, becoming concentrated predominantly in large aggregates. We assume that this coalescence (“patch” formation) is a necessary condition for lymphocyte triggering and briefly pursue the consequences.It is shown that the time required for patch formation is a sensitive function of affinity (K), antigen valence and antigen concentration (C), and that if KC is either too high or too low patch formation will not be possible. Moreover within the range of binding constants which can lead to patching, there will be an optimum value which leads to the fastest rate of triggering, and this optimum shifts to higher affinity as the concentration of free antigen surrounding the cell decreases. For optimum KC values we estimate times typically of the order of (10–100) seconds for patch formation. The theory also suggests that if antigen valence is too low, triggering will not be possible within times of interest, without introducing other factors. It thus leads naturally to a requirement for auxiliary cells which would tend to present low valence antigens in such a way that the B lymphocytes see an effective, increased valence. The theory, although primitive, thus meets some minimal requirements in that it distinguishes binding reactions from triggering reactions, makes predictions consistent with observations on affinity maturation and the nonresponsiveness to high doses and low doses of antigen, and suggests the need for helper cells (or their products) in order for low valence antigens to be effective in lymphocyte triggering.     

Folding and aggregation kinetics of a beta-hairpin

We have investigated the solution structure, equilibrium properties, and folding kinetics of a 17-residue beta-hairpin-forming peptide derived from the protein ubiquitin. NMR experiments show that at 4 degrees C the peptide has a highly populated beta-hairpin conformation. At protein concentrations higher than 0.35 mM, the peptide aggregates. Sedimentation equilibrium measurements show that the aggregate is a trimer, while NMR indicates that the beta-hairpin conformation is maintained in the trimer. The relaxation kinetics in nanosecond laser temperature-jump experiments reveal a concentration-independent microsecond phase, corresponding to beta-hairpin unfolding-refolding, and a concentration-dependent millisecond phase due to oligomerization. Kinetic modeling of the relaxation rates and amplitudes yields the folding and unfolding rates for the monomeric beta-hairpin, as well as assembly and disassembly rates for trimer formation consistent with the equilibrium constant determined by sedimentation equilibrium. When the net charge on the peptides and ionic strength were taken into account, the rate of trimer assembly approaches the Debye-Smoluchowski diffusion limit. At 300 K, the rate of formation of the monomeric hairpin is (17 micros)(-1), compared to rates of (0.8 micros)(-1) to (52 micros)(-1) found for other peptides. After using Kramers theory to correct for the temperature dependence of the pre-exponential factor, the activation energy for hairpin formation is near zero, indicating that the barrier to folding is purely entropic. Comparisons with previously measured rates for a series of hairpins are made to distinguish between zipper and hydrophobic collapse mechanisms. Overall, the experimental data are most consistent with the zipper mechanism in which structure formation is initiated at the turn, the mechanism predicted by the Ising-like statistical mechanical model that was developed to explain the equilibrium and kinetic data for the beta-hairpin from protein GB1. In contrast, the majority of simulation studies favor a hydrophobic collapse mechanism. However, with few exceptions, there is little or no quantitative comparison of the simulation results with experimental data.     

Interpreting the aggregation kinetics of amyloid peptides

Amyloid fibrils are insoluble mainly beta-sheet aggregates of proteins or peptides. The multi-step process of amyloid aggregation is one of the major research topics in structural biology and biophysics because of its relevance in protein misfolding diseases like Alzheimer's, Parkinson's, Creutzfeld-Jacob's, and type II diabetes. Yet, the detailed mechanism of oligomer formation and the influence of protein stability on the aggregation kinetics are still matters of debate. Here a coarse-grained model of an amphipathic polypeptide, characterized by a free energy profile with distinct amyloid-competent (i.e. beta-prone) and amyloid-protected states, is used to investigate the kinetics of aggregation and the pathways of fibril formation. The simulation results suggest that by simply increasing the relative stability of the beta-prone state of the polypeptide, disordered aggregation changes into fibrillogenesis with the presence of oligomeric on-pathway intermediates, and finally without intermediates in the case of a very stable beta-prone state. The minimal-size aggregate able to form a fibril is generated by collisions of oligomers or monomers for polypeptides with unstable or stable beta-prone state, respectively. The simulation results provide a basis for understanding the wide range of amyloid-aggregation mechanisms observed in peptides and proteins. Moreover, they allow us to interpret at a molecular level the much faster kinetics of assembly of a recently discovered functional amyloid with respect to the very slow pathological aggregation.     

Coarse-grained models for protein aggregation

The aggregation of soluble proteins into fibrillar species is a complex process that spans many lengths and time scales, and that involves the formation of numerous on-pathway and off-pathway intermediate species. Despite this complexity, several elements underlying the aggregation process appear to be universal. The kinetics typically follows a nucleation-growth process, and proteins with very different sequences aggregate to form similar fibril structures, populating intermediates with sufficient structural similarity to bind to a common antibody. This review focuses on a computational approach that exploits the common features of aggregation to simplify or 'coarse-grain' the representation of the protein. We highlight recent developments in coarse-grained modeling and illustrate how these models have been able to shed new light into the mechanisms of protein aggregation and the nature of aggregation intermediates. The roles of aggregation prone conformations in the monomeric state and the influence of inherent β-sheet and aggregation propensities in modulating aggregation pathways are discussed.     

Stochastic models for aggregation processes

Three models are presented, which describe the aggregation of objects into groups and the distributions of groups sizes and group numbers within habitats. The processes regarded are pure accumulation processes which involve only formation and invasion of groups. Invasion represents the special case of fusion when only single objects - and not groups - join a group of certain size. The basic model is derived by a single parameter, the formation probability q, which represents the probability of an object to form a new group. A novel, discrete and finite distribution that results for the group sizes is deduced from this aggregation process and it is shown that it converges to a geometric distribution if the number of objects tends to infinity. Two extensions of this model, which both converge to the Waring distribution, are added: the model can be extended either with a beta distributed formation probability or with the assumption that the invasion probability depends on the group size. Relationships between the limiting distributions involved are discussed.     

Sponge cell aggregation

Summary Dissociated sponge cell system has proved to be a useful model to study the process of cell aggregation both on cellular and subcellular level. The purpose of this review is to discuss recent results obtained from experiments with the marine sponge Geodia cydonium.Dissociated cells form functional aggregates during a process which can be sub-divided into three phases: first, formation of small primary aggregates in the presence of Ca²⁺; second, formation of secondary aggregates in the presence of an aggregation factor and third, reconstitution of a functional system of watercontaining channels by rearrangement in the secondary aggregates.On subcellular level a series of macromolecules are known which are involved in the control of aggregation and separation of sponge cells: Aggregation factor, aggregation receptor, anti-aggregation receptor, glucuronidase, ß-glucuronosyltransferase, ß- ß-galactosidase and a lectin. These components might be linked in the following sequence: (a) Activation of the aggregation receptor by its enzymic glucuronylation; (b) Adhesive recognition of the cells, mediated by the aggregation factor and the glucuronylated aggregation receptor; (c) Inactivation of the aggregation receptor by its deglucuronylation with the membrane-associated ß-glucuronidase; (d) Cell separation due to either the loss of the recognition site (glucuronic acid) of the aggregation receptor for the aggregation factor or to an inactivation of the aggregation factor by the anti-aggregation receptor. The activity of the anti-aggregation receptor is most likely controlled by the Geodia lectin.The events leading to cell-cell recognition cause a change in the following metabolic events: Increase of oxygen uptake, decrease of cyclic AMP level, increase of cyclic GMP level and stimulation of programmed syntheses.Abbreviations AF aggregation factor - CPP circular proteid particle - AR aggregation receptor - aAR anti-aggregation receptor - CMF calcium- and magnesium-free artificial sea water - ASW calcium- and magnesium-containing artificial sea water This paper is dedicated to PROF. DR. R. K. ZAHN on the occasion of his 60^th birthday.     

Viral aggregation: quantitation and kinetics of the aggregation of poliovirus and reovirus.

The aggregation of poliovirus and reovirus was followed in buffers at various pH values by means of a single particle analysis (SPA) test. The SPA test used here was modified from the original test reported earlier to prevent disaggregation of virus clumps from invalidating the results. The modified SPA test demonstrated that the efficiency of aggregation, which is a measure of the percentage of collisions which are effective in producing an aggregate, may vary widely depending on the conditions in which the virus is placed. The modified SPA test was also used to demonstrate that the kinetic features of viral aggregation follow the classical laws of colloid particle aggregation, which in turn are solely dependent upon diffusion of the particles as caused by brownian motion.     

The kinetics of aggregation of poly-glutamic acid based polypeptides

The aggregation of two negatively-charged polypeptides, poly-L-glutamic acid (PE) and a copolymer of poly-glutamic acid and poly-alanine (PEA), has been studied at different peptide and salt concentrations and solution pH conditions. The kinetics of aggregation were based on Thioflavin T (ThT) fluorescence measurements. The observed lag phase shortened and the aggregation was faster as the pH approached the polypeptides' isoelectric points. While the initial polypeptide structures of PE and PEA appeared identical as determined from circular dichroism spectroscopy, the final aggregate morphology differed; PE assumed large twisted lamellar structures and the PEA formed typical amyloid-like fibrils, although both contained extensive beta-sheet structure. Differences in aggregation behavior were observed for the two polypeptides as a function of salt concentration; aggregation progressed more slowly for PE and more quickly for PEA with increasing salt concentration. Several models of aggregation kinetics were fit to the data. No model yielded consistent rate constants or a critical nucleus size. A modified nucleated polymerization model was developed based on that of Powers and Powers [E.T. Powers, D.L. Powers, The kinetics of nucleated polymerizations at high concentrations: Amyloid fibril formation near and above the supercritical concentration, Biophys. J. 91 (2006) 122-132], which incorporated the ability of oligomeric species to interact. This provided a best fit to the experimental data.     

Minimal models of multi-site ligand-binding kinetics

Mathematical models based on the current understanding of co-operativity in ligand binding to the (macro) molecule and relating the dose-response (saturation) curve of the (macro) molecule ligation to intrinsic dissociation constants characterizing the affinities of ligand for binding sites of both unliganded and partly liganded (macro) molecule have been developed. The simplified models disregarding the structural properties and considerations concerning conformational changes of the (macro) molecule retain the ability to yield sigmoid curves of ligand binding and reflect the co-operativity. Model 1 contains only three parameters, parameter κ (a multiplier characterising the change in the affinity) reflects also the existence and type of co-operativity of ligand binding: κ<1 corresponds to positive co-operativity, κ>1 to the negative and κ=1 to the absence of any co-operativity. Model 2 contains an extra parameter, ω, equilibrium constant for the T₀↔R₀ transition but fails to produce dose-response, which would suggest negative co-operativity. For any fixed n>1, the deviation of the dose-response (saturation) curve from the Henri hyperbola depends either solely on parameter κ (Model 1) or also on parameter ω (Model 2). The (macro) molecule being a receptor, both models yield a diversity of dose-response curves due to possible variety of efficacies of the (macro) molecule. The models may be considered as extensions of the Henri model: in case the dissociation constants remain unchanged, the proposed models are reduced to the latter.     

Mass action kinetics of virus-cell aggregation and fusion.

A simple approximate solution for the mass action kinetics of small particles (viruses or vesicles) binding to large particles (cells) and their subsequent fusion has been derived. The solution is evaluated in terms of the measurable fluorescence changes expected when the virus or vesicles are labeled with fluorescent probes, which are diluted into the cellular membrane by fusion. Comparison with numerical integrations shows that the approximate solution is extremely accurate. Analytic simplifications for a variety of special cases of this general problem are also shown.     

Pseudophosphorylation of tau protein directly modulates its aggregation kinetics

Hyperphosphorylation of tau protein is associated with neurofibrillary lesion formation in Alzheimer's disease and other tauopathic neurodegenerative diseases. It fosters lesion formation by increasing the concentration of free tau available for aggregation and by directly modulating the tau aggregation reaction. To clarify how negative charge incorporation into tau directly affects aggregation behavior, the fibrillization of pseudophosphorylation mutant T212E prepared in a full-length four-repeat tau background was examined in vitro as a function of time and submicromolar tau concentrations using electron microscopy assay methods. Kinetic constants for nucleation and extension phases of aggregation were then estimated by direct measurement and mathematical simulation. Kinetic analysis revealed that pseudophosphorylation increased tau aggregation rate by increasing the rate of filament nucleation. In addition, it increased aggregation propensity by stabilizing mature filaments against disaggregation. The data suggest that incorporation of negative charge into the T212 site can directly promote tau filament formation at multiple steps in the aggregation pathway.     

The kinetics of cellular aggregation induced by turbulent flow

A theoretical treatment for the aggregation of cellular dispersions in a turbulent fluid is proposed based on the work of Saffman &; Turner (1956). The use of an expression for the rate of collisions between cells and cell aggregates which is dependent on the size of the colliding cell particles gives theoretical results which markedly reflect many of the features of cellular aggregation as found experimentally by pulse height analysis using a Coulter counter and particle size discriminator. In particular the shape of the distribution curves, the rate of change of single cell population and the attainment of an equilibrium state as well as the occurrence of cell aggregate redistribution during aggregation are shown to be consistent with aggregation in a turbulent field.It is also shown that the nature of the initial cell aggregate distribution has a very significant effect on subsequent aggregation kinetics.The theory has been applied to the aggregation of two Chinese hamster cell lines and gives a satisfactory explanation of the experimental results.