Stochastic self-assembly of incommensurate clusters

Theoretical and computational methodologies for understanding coordination self-assembly complexes

This perspective highlights three theoretical and computational methods to capture the coordination self-assembly processes at the molecular level: quantum chemical modeling, molecular dynamics, and reaction network analysis. These methods cover the different scales from the metal-ligand bond to a more global aspect, and approaches that are best suited to understand the coordination self-assembly from different perspectives are introduced. Theoretical and numerical researches based on these methods are not merely ways of interpreting the experimental studies but complementary to them.

Optimizing dynamical functions for speed with stochastic paths

Living systems are built from microscopic components that function dynamically; they generate work with molecular motors, assemble and disassemble structures such as microtubules, keep time with circadian clocks, and catalyze the replication of DNA. How do we implement these functions in synthetic nanostructured materials to execute them before the onset of dissipative losses? Answering this question requires a quantitative understanding of when we can improve performance and speed while minimizing the dissipative losses associated with operating in a fluctuating environment. Here, we show that there are four modalities for optimizing dynamical functions that can guide the design of nanoscale systems. We analyze Markov models that span the design space: a clock, ratchet, replicator, and self-assembling system. Using stochastic thermodynamics and an exact expression for path probabilities, we classify these models of dynamical functions based on the correlation of speed with dissipation and with the chosen performance metric. We also analyze random networks to identify the model features that affect their classification and the optimization of their functionality. Overall, our results show that the possible nonequilibrium paths can determine our ability to optimize the performance of dynamical functions, despite ever-present dissipation, when there is a need for speed.

Stochastic paths controlling speed and dissipation

Natural processes occur in a finite amount of time and dissipate energy, entropy, and matter. Near equilibrium, thermodynamic intuition suggests that fast irreversible processes will dissipate more energy and entropy than slow quasistatic processes connecting the same initial and final states. For small systems, recently discovered thermodynamic speed limits suggest that faster processes will dissipate more than slower processes. Here, we test the hypothesis that this relationship between speed and dissipation holds for stochastic paths far from equilibrium. To analyze stochastic paths on finite timescales, we derive an exact expression for the path probabilities of continuous-time Markov chains from the path summation solution to the master equation. We present a minimal model for a driven system in which relative energies of the initial and target states control the speed, and the nonequilibrium currents of a cycle control the dissipation. Although the hypothesis holds near equilibrium, we find that faster processes can dissipate less under far-from-equilibrium conditions because of strong currents. This model serves as a minimal prototype for designing kinetics to sculpt the nonequilibrium path space so that faster paths produce less dissipation.

Discrete stochastic models of SELEX: Aptamer capture probabilities and protocol optimization

Antibodies are important biomolecules that are often designed to recognize target antigens. However, they are expensive to produce and their relatively large size prevents their transport across lipid membranes. An alternative to antibodies is aptamers, short ([Formula: see text] bp) oligonucleotides (and amino acid sequences) with specific secondary and tertiary structures that govern their affinity to specific target molecules. Aptamers are typically generated via solid phase oligonucleotide synthesis before selection and amplification through Systematic Evolution of Ligands by EXponential enrichment (SELEX), a process based on competitive binding that enriches the population of certain strands while removing unwanted sequences, yielding aptamers with high specificity and affinity to a target molecule. Mathematical analyses of SELEX have been formulated in the mass action limit, which assumes large system sizes and/or high aptamer and target molecule concentrations. In this paper, we develop a fully discrete stochastic model of SELEX. While converging to a mass-action model in the large system-size limit, our stochastic model allows us to study statistical quantities when the system size is small, such as the probability of losing the best-binding aptamer during each round of selection. Specifically, we find that optimal SELEX protocols in the stochastic model differ from those predicted by a deterministic model.

Diversity in biology: definitions, quantification and models

Diversity indices are useful single-number metrics for characterizing a complex distribution of a set of attributes across a population of interest. The utility of these different metrics or sets of metrics depends on the context and application, and whether a predictive mechanistic model exists. In this topical review, we first summarize the relevant mathematical principles underlying heterogeneity in a large population, before outlining the various definitions of 'diversity' and providing examples of scientific topics in which its quantification plays an important role. We then review how diversity has been a ubiquitous concept across multiple fields, including ecology, immunology, cellular barcoding experiments, and socioeconomic studies. Since many of these applications involve sampling of populations, we also review how diversity in small samples is related to the diversity in the entire population. Features that arise in each of these applications are highlighted.

Stochastic yield catastrophes and robustness in self-assembly

A guiding principle in self-assembly is that, for high production yield, nucleation of structures must be significantly slower than their growth. However, details of the mechanism that impedes nucleation are broadly considered irrelevant. Here, we analyze self-assembly into finite-sized target structures employing mathematical modeling. We investigate two key scenarios to delay nucleation: (i) by introducing a slow activation step for the assembling constituents and, (ii) by decreasing the dimerization rate. These scenarios have widely different characteristics. While the dimerization scenario exhibits robust behavior, the activation scenario is highly sensitive to demographic fluctuations. These demographic fluctuations ultimately disfavor growth compared to nucleation and can suppress yield completely. The occurrence of this stochastic yield catastrophe does not depend on model details but is generic as soon as number fluctuations between constituents are taken into account. On a broader perspective, our results reveal that stochasticity is an important limiting factor for self-assembly and that the specific implementation of the nucleation process plays a significant role in determining the yield.

The self-assembly of a large biological molecule from small building blocks is like finishing a puzzle of magnetic pieces by shaking the box. Even though each piece of the puzzle is attracted to its correct neighbours, the limited control makes it very hard to finish the puzzle in a short amount of time.

The problem becomes even more difficult if several copies of the same puzzle are assembled in one box. If several puzzles start at the same time, the different parts might steal pieces from each other, making it impossible to successfully complete any of the puzzles. This is called a depletion trap. If the box is only shaken and there is no real control over individual pieces, these traps occur at random.

Overcoming these random depletion traps is an important challenge when assembling nanostructures and other artificial molecules designed by humans without wasting many, potentially expensive, components. Previous studies have shown that when multiple copies of the same structure are assembled simultaneously, slowing the rate of initiation increases the yield of correctly-made structures. This prevents new structures from stealing pieces from existing structures before they are fully completed.

Now, Gartner, Graf, Wilke et al. have used a mathematical model to show that changing the way initiation is delayed leads to different yields. This was especially true for small systems where fluctuations in the availability of the different pieces strongly enhanced the initiation of new structures. In these cases, the self-assembly process terminated undesirably with many incomplete structures.

Nanostructures have various applications ranging from drug delivery to robotics. These findings suggest that in order to efficiently assemble biological molecules, the concentrations of the different building blocks need to be tightly controlled. A question for further research is to investigate strategies that reduce fluctuations in the availability of the building blocks to develop more efficient assembly protocols.

A kinetics study of ligand substitution reaction on dinuclear platinum complexes: Stochastic versus deterministic approach

The kinetics on a basic ligand substitution reaction on dinuclear platinum complexes [Pt(PEt₃ )₂ PhPt(PEt₃ )₂ ]²⁺ and [Pt(PEt₃ )₂ PhCOPhPt(PEt₃ )₂ ]²⁺ , with the ligands pyridine and 3-chloropyridine, is studied. This is a fundamental step in a self-assembly, and the time evolution has been observed with a new experimental technique, QASAP (quantitative analysis of self-assembly process), which is recently developed by Hiraoka's group. As a result of numerical calculations based on master equation, we succeed in specifying the reaction rate constants with a simple reaction model. In addition, the time evolutions of all the intermediate components produced and consumed in chemical reaction are revealed, including those unobserved in the experiments. The convergence behavior of the existence ratios of specific chemical species calculated with the stochastic algorithm method is compared with those obtained from deterministic formalism based on rate equations, revealing a clear dependence on the number of constituent molecules. © 2018 Wiley Periodicals, Inc.

Evolution of Coagulation-Fragmentation Stochastic Processes Using Accurate Chemical Master Equation Approach

Coagulation and fragmentation (CF) is a fundamental process in which smaller particles attach to each other to form larger clusters while existing clusters break up into smaller particles . It is a ubiquitous process that plays important roles in many physical and biological phenomena. CF is typically a stochastic process that often occurs in confined spaces with a limited number of available particles . Here, we study the CF process formulated with the discrete Chemical Master Equation (dCME). Using the newly developed Accurate Chemical Master Equation (ACME) method, we examine the time-dependent behavior of the CF system. We investigate the effects of a number of important factors that influence the overall behavior of the system, including the dimensionality, the ratio of attachment to detachment rates among clusters, and the initial conditions. By comparing CF in one and three dimensions, we conclude that systems in three dimensions are more likely to form large clusters. We also demonstrate how the ratio of the attachment to detachment rates affects the dynamics and the steady-state of the system. Finally, we demonstrate the relationship between the formation of large clusters and the initial condition.

Generalizing a mathematical model of prion aggregation allows strain coexistence and co-stability by including a novel misfolded species

Prions are proteins capable of adopting misfolded conformations and transmitting these conformations to other normally folded proteins. Prions are most commonly known for causing fatal neurodegenerative diseases in mammals but are also associated with several harmless phenotypes in yeast. A distinct feature of prion propagation is the existence of different phenotypical variants, called strains. It is widely accepted that these strains correspond to different conformational states of the protein, but the mechanisms driving their interactions remain poorly understood. This study uses mathematical modeling to provide insight into this problem. We show that the classical model of prion dynamics allows at most one conformational strain to stably propagate. In order to conform to biological observations of strain coexistence and co-stability, we develop an extension of the classical model by introducing a novel prion species consistent with biological studies. Qualitative analysis of this model reveals a new variety of behavior. Indeed, it allows for stable coexistence of different strains in a wide parameter range, and it also introduces intricate initial condition dependency. These new behaviors are consistent with experimental observations of prions in both mammals and yeast. As such, our model provides a valuable tool for investigating the underlying mechanisms of prion propagation and the link between prion strains and strain specific phenotypes. The consideration of a novel prion species brings a change in perspective on prion biology and we use our model to generate hypotheses about prion infectivity.

Regulatory inhibition of biological tissue mineralization by calcium phosphate through post-nucleation shielding by fetuin-A

In vertebrates, insufficient availability of calcium and inorganic phosphate ions in extracellular fluids leads to loss of bone density and neuronal hyper-excitability. To counteract this problem, calcium ions are usually present at high concentrations throughout bodily fluids-at concentrations exceeding the saturation point. This condition leads to the opposite situation where unwanted mineral sedimentation may occur. Remarkably, ectopic or out-of-place sedimentation into soft tissues is rare, in spite of the thermodynamic driving factors. This fortunate fact is due to the presence of auto-regulatory proteins that are found in abundance in bodily fluids. Yet, many important inflammatory disorders such as atherosclerosis and osteoarthritis are associated with this undesired calcification. Hence, it is important to gain an understanding of the regulatory process and the conditions under which it can go awry. In this manuscript, we extend mean-field continuum classical nucleationtheory of the growth of clusters to encompass surface shielding. We use this formulation to study the regulation of sedimentation of calcium phosphate salts in biological tissues through the mechanism of post-nuclear shielding of nascent mineral particles by binding proteins. We develop a mathematical description of this phenomenon using a countable system of hyperbolic partial differential equations. A critical concentration of regulatory protein is identified as a function of the physical parameters that describe the system.

Cooperative Clustering Digitizes Biochemical Signaling and Enhances its Fidelity

Many membrane-bound molecules in cells form small clusters. It has been hypothesized that these clusters convert an analog extracellular signal into a digital intracellular signal and that this conversion increases signaling fidelity. However, the mechanism by which clusters digitize a signal and the subsequent effects on fidelity remain poorly understood. Here we demonstrate using a stochastic model of cooperative cluster formation that sufficient cooperation leads to digital signaling. We show that despite reducing the number of output states, which decreases fidelity, digitization also reduces noise in the system, which increases fidelity. The tradeoff between these effects leads to an optimal cluster size that agrees with experimental measurements.

Modeling the Kinetics of Open Self-Assembly

In this work, we explore theoretically the kinetics of molecular self-assembly in the presence of constant monomer flux as an input, and a maximal size. The proposed model is supposed to reproduce the dynamics of viral self-assembly for enveloped virus. It turns out that the kinetics of open self-assembly is rather quantitatively different from the kinetics of similar closed assembly. In particular, our results show that the convergence toward the stationary state is reached through assembly waves. Interestingly, we show that the production of complete clusters is much more efficient in the presence of a constant input flux, rather than providing all monomers at the beginning of the self-assembly.

First passage times in homogeneous nucleation: Dependence on the total number of particles

Motivated by nucleation and molecular aggregation in physical, chemical, and biological settings, we present an extension to a thorough analysis of the stochastic self-assembly of a fixed number of identical particles in a finite volume. We study the statistics of times required for maximal clusters to be completed, starting from a pure-monomeric particle configuration. For finite volumes, we extend previous analytical approaches to the case of arbitrary size-dependent aggregation and fragmentation kinetic rates. For larger volumes, we develop a scaling framework to study the first assembly time behavior as a function of the total quantity of particles. We find that the mean time to first completion of a maximum-sized cluster may have a surprisingly weak dependence on the total number of particles. We highlight how higher statistics (variance, distribution) of the first passage time may nevertheless help to infer key parameters, such as the size of the maximum cluster. Finally, we present a framework to quantify formation of macroscopic sized clusters, which are (asymptotically) very unlikely and occur as a large deviation phenomenon from the mean-field limit. We argue that this framework is suitable to describe phase transition phenomena, as inherent infrequent stochastic processes, in contrast to classical nucleation theory.

Initial condition of stochastic self-assembly

The formation of a stable protein aggregate is regarded as the rate limiting step in the establishment of prion diseases. In these systems, once aggregates reach a critical size the growth process accelerates and thus the waiting time until the appearance of the first critically sized aggregate is a key determinant of disease onset. In addition to prion diseases, aggregation and nucleation is a central step of many physical, chemical, and biological process. Previous studies have examined the first-arrival time at a critical nucleus size during homogeneous self-assembly under the assumption that at time t=0 the system was in the all-monomer state. However, in order to compare to in vivo biological experiments where protein constituents inherited by a newly born cell likely contain intermediate aggregates, other possibilities must be considered. We consider one such possibility by conditioning the unique ergodic size distribution on subcritical aggregate sizes; this least-informed distribution is then used as an initial condition. We make the claim that this initial condition carries fewer assumptions than an all-monomer one and verify that it can yield significantly different averaged waiting times relative to the all-monomer condition under various models of assembly.

Hidden entropic contribution in the thermodynamics of molecular complexation

It has become an axiom that the thermodynamic analysis of non-covalent molecular complexation is intrinsically model-dependent, i.e. the set of implicitly or explicitly introduced assumptions may strongly affect the thermodynamic parameters.

Mechanisms of blood homeostasis: lineage tracking and a neutral model of cell populations in rhesus macaques

Background

How a potentially diverse population of hematopoietic stem cells (HSCs) differentiates and proliferates to supply more than 10¹¹ mature blood cells every day in humans remains a key biological question. We investigated this process by quantitatively analyzing the clonal structure of peripheral blood that is generated by a population of transplanted lentivirus-marked HSCs in myeloablated rhesus macaques. Each transplanted HSC generates a clonal lineage of cells in the peripheral blood that is then detected and quantified through deep sequencing of the viral vector integration sites (VIS) common within each lineage. This approach allowed us to observe, over a period of 4-12 years, hundreds of distinct clonal lineages.

Results

While the distinct clone sizes varied by three orders of magnitude, we found that collectively, they form a steady-state clone size-distribution with a distinctive shape. Steady-state solutions of our model show that the predicted clone size-distribution is sensitive to only two combinations of parameters. By fitting the measured clone size-distributions to our mechanistic model, we estimate both the effective HSC differentiation rate and the number of active HSCs.

Conclusions

Our concise mathematical model shows how slow HSC differentiation followed by fast progenitor growth can be responsible for the observed broad clone size-distribution. Although all cells are assumed to be statistically identical, analogous to a neutral theory for the different clone lineages, our mathematical approach captures the intrinsic variability in the times to HSC differentiation after transplantation.

Electronic supplementary material

The online version of this article (doi:10.1186/s12915-015-0191-8) contains supplementary material, which is available to authorized users.

Inherent variability in the kinetics of autocatalytic protein self-assembly

In small volumes, the kinetics of filamentous protein self-assembly is expected to show significant variability, arising from intrinsic molecular noise. This is not accounted for in existing deterministic models. We introduce a simple stochastic model including nucleation and autocatalytic growth via elongation and fragmentation, which allows us to predict the effects of molecular noise on the kinetics of autocatalytic self-assembly. We derive an analytic expression for the lag-time distribution, which agrees well with experimental results for the fibrillation of bovine insulin. Our expression decomposes the lag-time variability into contributions from primary nucleation and autocatalytic growth and reveals how each of these scales with the key kinetic parameters. Our analysis shows that significant lag-time variability can arise from both primary nucleation and from autocatalytic growth and should provide a way to extract mechanistic information on early-stage aggregation from small-volume experiments.

Combinatoric analysis of heterogeneous stochastic self-assembly

We analyze a fully stochastic model of heterogeneous nucleation and self-assembly in a closed system with a fixed total particle number M, and a fixed number of seeds Ns. Each seed can bind a maximum of N particles. A discrete master equation for the probability distribution of the cluster sizes is derived and the corresponding cluster concentrations are found using kinetic Monte-Carlo simulations in terms of the density of seeds, the total mass, and the maximum cluster size. In the limit of slow detachment, we also find new analytic expressions and recursion relations for the cluster densities at intermediate times and at equilibrium. Our analytic and numerical findings are compared with those obtained from classical mass-action equations and the discrepancies between the two approaches analyzed.

Computer simulation of assembly and co-operativity of hexameric AAA ATPases

AAA ATPases form a functionally diverse superfamily of proteins. Most members form homo-hexameric ring complexes, are catalytically active only in the fully assembled state, and show co-operativity among the six subunits. The mutual dependence among the subunits is clearly evidenced by the fact that incorporation of mutated, inactive subunits can decrease the activity of the remaining wild type subunits. For the first time, we develop here models to describe this form of allostery, evaluate them in a simulation study, and test them on experimental data. We show that it is important to consider the assembly reactions in the kinetic model, and to define a formal inhibition scheme. We simulate three inhibition scenarios explicitly, and demonstrate that they result in differing outcomes. Finally, we deduce fitting formulas, and test them on real and simulated data. A non-competitive inhibition formula fitted experimental and simulated data best. To our knowledge, our study is the first one that derives and tests formal allosteric schemes to explain the inhibitory effects of mutant subunits on oligomeric enzymes.

First passage times in homogeneous nucleation and self-assembly

Motivated by nucleation and molecular aggregation in physical, chemical, and biological settings, we present a thorough analysis of the general problem of stochastic self-assembly of a fixed number of identical particles in a finite volume. We derive the backward Kolmogorov equation (BKE) for the cluster probability distribution. From the BKE, we study the distribution of times it takes for a single maximal cluster to be completed, starting from any initial particle configuration. In the limits of slow and fast self-assembly, we develop analytical approaches to calculate the mean cluster formation time and to estimate the first assembly time distribution. We find, both analytically and numerically, that faster detachment can lead to a shorter mean time to first completion of a maximum-sized cluster. This unexpected effect arises from a redistribution of trajectory weights such that upon increasing the detachment rate, paths that take a shorter time to complete a cluster become more likely.

Random versus sequential pathway of molecular self-assembly

Two important assumptions are often made in the analysis of molecular self-assembly at equilibrium, viz., that sequential is preferred to random aggregation and that the equilibrium constants at each stage of aggregation are equal, though both assumptions have not been justified strictly. In the present work we show that molecular self-assembly leading to formation of linear polymers and proceeding in a random manner appears to be less entropically favored than sequential aggregation, which provides a physical background for assuming sequential aggregation when studying molecular self-assembly in solution. Exact equations for analysis of experimental data for molecular assembly proceeding in a sequential manner were derived by taking strict account of the profile of the equilibrium constant, which provides a physically more correct approach than that using the conventional indefinite equilibrium constant (EK) model.