On the theory of crystal growth in metastable systems with biomedical applications: protein and insulin crystallization

An experimental instrument for solidification and in situ characterization of the nucleation behavior of high-melting metal under a high magnetic field

A specially designed experimental apparatus suitable for commercial superconductor magnet is used for solidification and in situ characterization of the nucleation behavior of high-melting metals. In order to carry out solidification experiments under a high magnetic field (HMF), the sample cell in the experimental device has two stations for repeated verification experiments of two same samples or comparative experiments of two different samples. Meanwhile, a metal specimen and a reference (α-Al2O3) are placed in the sample cell to characterize the nucleation behavior in situ. Using this experimental device, the nucleation behaviors of Al-7wt. %Si alloy and pure Cu under a HMF were investigated. The results show that the undercoolings of Al-7wt. %Si alloy and pure Cu increase under the HMF. Furthermore, the applied HMF decreases the activation energy of Al-7wt. %Si alloy and increases the nucleation work. Based on the magnetohydrodynamic effect, the change in undercooling and nucleation work could be partly attributed to the restrained thermal convection by the HMF in this study.

Structural principles of insulin formulation and analog design: A century of innovation

BACKGROUND

The discovery of insulin in 1921 and its near-immediate clinical use initiated a century of innovation. Advances extended across a broad front, from the stabilization of animal insulin formulations to the frontiers of synthetic peptide chemistry, and in turn, from the advent of recombinant DNA manufacturing to structure-based protein analog design. In each case, a creative interplay was observed between pharmaceutical applications and then-emerging principles of protein science; indeed, translational objectives contributed to a growing molecular understanding of protein structure, aggregation and misfolding.

SCOPE OF REVIEW

Pioneering crystallographic analyses-beginning with Hodgkin's solving of the 2-Zn insulin hexamer-elucidated general features of protein self-assembly, including zinc coordination and the allosteric transmission of conformational change. Crystallization of insulin was exploited both as a step in manufacturing and as a means of obtaining protracted action. Forty years ago, the confluence of recombinant human insulin with techniques for site-directed mutagenesis initiated the present era of insulin analogs. Variant or modified insulins were developed that exhibit improved prandial or basal pharmacokinetic (PK) properties. Encouraged by clinical trials demonstrating the long-term importance of glycemic control, regimens based on such analogs sought to resemble daily patterns of endogenous β-cell secretion more closely, ideally with reduced risk of hypoglycemia.

MAJOR CONCLUSIONS

Next-generation insulin analog design seeks to explore new frontiers, including glucose-responsive insulins, organ-selective analogs and biased agonists tailored to address yet-unmet clinical needs. In the coming decade, we envision ever more powerful scientific synergies at the interface of structural biology, molecular physiology and therapeutics.

Evaporation kinetics of a polydisperse ensemble of drops

A mathematical model of the evaporation of a polydisperse ensemble of drops, with allowance for a nonlinear 'diffusion' term in the kinetic equation for the population density distribution function, is developed. The model describes the interaction of a gas phase with vaporizing drops: it has great potential for application in condensed matter physics, thermophysics and engineering devices (e.g. spray drying, cooling, power engineering). The kinetics of heat transfer between phases is theoretically studied. An analytical solution to the integro-differential equations of the process of droplet evaporation is found in a parametric form. Analytical solutions in the presence and absence of the 'diffusion' term are compared. It is shown that the fluctuations in particle evaporation rates ('diffusion' term in the Fokker-Planck equation) play a decisive role in the evolutionary behaviour of a polydisperse ensemble of vaporizing liquid drops. This article is part of the theme issue 'Transport phenomena in complex systems (part 1)'.

The influence of non-stationarity and interphase curvature on the growth dynamics of spherical crystals in a metastable liquid

This manuscript is concerned with the theory of nucleation and evolution of a polydisperse ensemble of crystals in metastable liquids during the intermediate stage of a phase transformation process. A generalized growth rate of individual crystals is obtained with allowance for the effects of their non-stationary evolution in unsteady temperature (solute concentration) field and the phase transition temperature shift appearing due to the particle curvature (the Gibbs-Thomson effect) and atomic kinetics. A complete system of balance and kinetic equations determining the transient behaviour of the metastability degree and the particle-radius distribution function is analytically solved in a parametric form. The coefficient of mutual Brownian diffusion in the Fokker-Planck equation is considered in a generalized form defined by an Einstein relation. It is shown that the effects under consideration substantially change the desupercooling/desupersaturation dynamics and the transient behaviour of the particle-size distribution function. The asymptotic state of the distribution function (its 'tail'), which determines the relaxation dynamics of the concluding (Ostwald ripening) stage of a phase transformation process, is derived. This article is part of the theme issue 'Transport phenomena in complex systems (part 1)'.

Nucleation and growth dynamics of ellipsoidal crystals in metastable liquids

When describing the growth of crystal ensembles from metastable solutions or melts, a significant deviation from a spherical shape is often observed. Experimental data show that the shape of growing crystals can often be considered ellipsoidal. The new theoretical models describing the transient nucleation of ellipsoidal particles and their growth with and without fluctuating rates at the intermediate stage of bulk phase transitions in metastable systems are considered. The nonlinear transport (diffusivity) of ellipsoidal crystals in the space of their volumes is taken into account in the Fokker-Planck equation allowing for fluctuating growth rates. The complete analytical solutions of integro-differential models of kinetic and balance equations are found and analysed. Our solutions show that the desupercooling dynamics is several times faster for ellipsoidal crystals as compared to spherical particles. In addition, the crystal-volume distribution function is lower and shifted to larger particle volumes when considering the growth of ellipsoidal crystals. What is more, this function is monotonically increasing to the maximum crystal size in the absence of fluctuations and is a bell-shaped curve when such fluctuations are taken into account. This article is part of the theme issue 'Transport phenomena in complex systems (part 1)'.

Ostwald ripening in the presence of simultaneous occurrence of various mass transfer mechanisms: an extension of the Lifshitz-Slyozov theory

The Ostwald ripening stage of a phase transformation process with allowance for synchronous operation of various mass transfer mechanisms (volume diffusion and diffusion along the block boundaries and dislocations) and the initial condition for the particle-radius distribution function is theoretically studied. The initial condition is taken from the analytical solution describing the intermediate stage of a phase transition process. The present theory focuses on relaxation dynamics from the beginning of the ripening process to its final asymptotic state, which is described by the previously constructed theories (Slezov VV. et al. 1978 J. Phys. Chem. Solids 39, 705-709. (doi:10.1016/0022-3697(78)90002-1) and Alexandrov & Alexandrova 2020 Phil. Trans. R. Soc. A 378, 20190247. (doi:10.1098/rsta.2019.0247)). An evolutionary behaviour of particle growth rates dependent on various mass transfer mechanisms and time is analytically described. The boundaries of the transition layer, which surround the blocking point, are found. The fundamental and relaxation contributions to the particle-radius distribution function are derived for the simultaneous occurrence of various mass transfer mechanisms. The left branch of this function is shifted to smaller particle radii whereas its right branch extends to the right of the blocking point as compared with the asymptotic universal distribution function. The theory under consideration well agrees with experimental data. This article is part of the theme issue 'Transport phenomena in complex systems (part 1)'.

Analytical solution of a binary melt solidification model in the presence of a quasi-equilibrium mushy region for the case of the non-linear phase diagram

The process of directional crystallization with a quasi-equilibrium mushy region is considered in the steady-state conditions with allowance for the non-linear phase diagram. A complete analytical solution of model equations is found in a parametric form by introducing a new variable-the solid phase fraction in the mushy layer. The temperature and solute concentration, as well as the solid fraction in the mushy layer, its boundaries, and the crystallization velocity, are determined analytically. It is shown that a non-linear phase diagram substantially influences the behavior of solutions obtained.

Characterization of amyloid β fibril formation under microgravity conditions

Amyloid fibrils are self-assembled and ordered proteinaceous supramolecules structurally characterized by the cross-β spine. Amyloid formation is known to be related to various diseases typified by neurogenerative disorders and involved in a variety of functional roles. Whereas common mechanisms for amyloid formation have been postulated across diverse systems, the mesoscopic morphology of the fibrils is significantly affected by the type of solution condition in which it grows. Amyloid formation is also thought to share a phenomenological similarity with protein crystallization. Although many studies have demonstrated the effect of gravity on protein crystallization, its effect on amyloid formation has not been reported. In this study, we conducted an experiment at the International Space Station (ISS) to characterize fibril formation of 40-residue amyloid β (Aβ(1-40)) under microgravity conditions. Our comparative analyses revealed that the Aβ(1-40) fibrilization progresses much more slowly on the ISS than on the ground, similarly to protein crystallization. Furthermore, microgravity promoted the formation of distinct morphologies of Aβ(1-40) fibrils. Our findings demonstrate that the ISS provides an ideal experimental environment for detailed investigations of amyloid formation mechanisms by eliminating the conventionally uncontrollable factors derived from gravity.

Dynamics of particulate assemblages in metastable liquids: a test of theory with nucleation and growth kinetics

This manuscript is devoted to the nonlinear dynamics of particulate assemblages in metastable liquids, caused by various dynamical laws of crystal growth and nucleation kinetics. First of all, we compare the quasi-steady-state and unsteady-state growth rates of spherical crystals in supercooled and supersaturated liquids. It is demonstrated that the unsteady-state rates transform to the steady-state ones in a limiting case of fine particles. We show that the real crystals evolve slowly in a more actual case of unsteady-state growth laws. Various growth rates of particles are tested against experimental data in metastable liquids. It is demonstrated that the unsteady-state rates describe the nonlinear behaviour of experimental curves with increasing the growth time or supersaturation. Taking this into account, the crystal-size distribution function and metastability degree are analytically found and compared with experimental data on crystallization in inorganic and organic solutions. It is significant that the distribution function is shifted to smaller sizes of particles if we are dealing with the unsteady-state growth rates. In addition, a complete analytical solution constructed in a parametric form is simplified in the case of small fluctuations in particle growth rates. In this case, a desupercooling/desupersaturation law is derived in an explicit form. Special attention is devoted to the biomedical applications for insulin and protein crystallization. This article is part of the theme issue 'Patterns in soft and biological matters'.

The effect of density changes on crystallization with a mushy layer

A nonlinear problem with two moving boundaries of the phase transition, which describes the process of directional crystallization in the presence of a quasi-equilibrium two-phase layer, is solved analytically for the steady-state process. The exact analytical solution in a two-phase layer is found in a parametric form (the solid phase fraction plays the role of this parameter) with allowance for possible changes in the density of the liquid phase accordingly to a linearized equation of state and arbitrary value of the solid fraction at the boundary between the two-phase and solid layers. Namely, the solute concentration, temperature, solid fraction in the mushy layer, liquid and solid phases, mushy layer thickness and its velocity are found analytically. The theory under consideration is in good agreement with experimental data. The obtained solutions have great potential applications in analysing similar processes with a two-phase layer met in materials science, geophysics, biophysics and medical physics, where the directional crystallization processes with a quasi-equilibrium mushy layer can occur. This article is part of the theme issue 'Patterns in soft and biological matters'.

From nucleation and coarsening to coalescence in metastable liquids

The transition of a metastable liquid (supersaturated solution or supercooled melt) occurring from the intermediate stage (where the crystals nucleate and grow) to the concluding stage (where the larger particles evolve at the expense of the dissolution of smaller particles) is theoretically described, with allowance for various mass transfer mechanisms (reaction on the interface surface, volume diffusion, grain-boundary diffusion, diffusion along the dislocations) arising at the stage of Ostwald ripening (coalescence). The initial distribution function (its 'tail') for the concluding stage (forming as a result of the evolution of a particulate assemblage during the intermediate stage) is taken into account to determine the particle-size distribution function at the stage of Ostwald ripening. This modified distribution function essentially differs from the universal Lifshitz-Slyozov (LS) solutions for several mass transfer mechanisms. Namely, its maximum lies below and is shifted to the left in comparison with the LS asymptotic distribution function. In addition, the right branch of the particle-size distribution lies above and is shifted to the right of the LS blocking point. It is shown that the initial 'tail' of the particle-size distribution function completely determines its behaviour at the concluding stage of Ostwald ripening. The present theory agrees well with experimental data. This article is part of the theme issue 'Patterns in soft and biological matters'.

Dissolution of polydisperse ensembles of crystals in channels with a forced flow

A non-stationary integro-differential model describing the dissolution of polydisperse ensembles of crystals in channels filled with flowing liquid is analysed. The particle-size distribution function, the particle flux through an arbitrary cross-section of the channel, the particle concentration profile, as well as the disappearance intensity of particles are found analytically. It is shown that a nonlinear behaviour of solutions is completely defined by the source term of particles introduced into the channel. In particular, the model approximately describes the processes of dissolution and transport of drug microcrystals to the target sites in a living organism, taking into account complex dissolution kinetics of drug particles. This article is part of the theme issue 'Patterns in soft and biological matters'.

Patterns in soft and biological matters

The issue is devoted to theoretical, computer and experimental studies of internal heterogeneous patterns, their morphology and evolution in various soft physical systems-organic and inorganic materials (e.g. alloys, polymers, cell cultures, biological tissues as well as metastable and composite materials). The importance of these studies is determined by the significant role of internal structures on the macroscopic properties and behaviour of natural and manufactured tissues and materials. Modern methods of computer modelling, statistical physics, heat and mass transfer, statistical hydrodynamics, nonlinear dynamics and experimental methods are presented and discussed. Non-equilibrium patterns which appear during macroscopic transport and hydrodynamic flow, chemical reactions, external physical fields (magnetic, electrical, thermal and hydrodynamic) and the impact of external noise on pattern evolution are the foci of this issue. Special attention is paid to pattern formation in biological systems (such as drug transport, hydrodynamic patterns in blood and pattern dynamics in protein and insulin crystals) and to the development of a scientific background for progressive methods of cancer and insult therapy (magnetic hyperthermia for cancer therapy; magnetically induced drug delivery in thrombosed blood vessels). The present issue includes works on pattern growth and their evolution in systems with complex internal structures, including stochastic dynamics, and the influence of internal structures on the external static, dynamic magnetic and mechanical properties of these systems. This article is part of the theme issue 'Patterns in soft and biological matters'.

On the theory of the unsteady-state growth of spherical crystals in metastable liquids

Motivated by a large number of applications, we consider the process of non-stationary growth of spherical crystals in a supercooled binary melt. The moving-boundary problem describing the unsteady-state distributions of temperature and impurity concentration around the growing crystal as well as the dynamics of its radius and growth rate is solved by means of the methods of small-parameter expansion and Laplace-Carson integral transform. We show that the growth rate of crystals contains the main contribution (which is proportional to the supercooling degree Δ) and the first correction (which is proportional to Δ² t, where t is time). The second correction is also found. The non-stationary temperature and concentration fields are determined as power functions of Δ and t. We demonstrate that the first corrections to the dynamics of crystal radius R( t) and its growth rate V ( t) play an important role. It is shown that R( t) and V ( t) can change more than twice in comparison with the previously known steady-state solution with the course of time. Such a behaviour will significantly modify the dynamics of a polydisperse ensemble of crystals evolving in a metastable liquid. This article is part of the theme issue 'Heterogeneous materials: metastable and non-ergodic internal structures'.

Effects of nonlinear growth rates of spherical crystals and their withdrawal rate from a crystallizer on the particle-size distribution function

In this paper, we show that the nonlinear growth rate of particles in a supersaturated solution or supercooled melt, as well as the rate of removal of crystals from the metastable liquid of a crystallizer, significantly change the size-distribution function of crystals. Taking these rates into account, we present a complete analytical solution of the integro-differential model describing the transient nucleation of solid particles and their evolution in a metastable liquid. The distribution function and metastability degree (supersaturation or supercooling) are found by means of the separation of variables and saddle-point methods. The nonlinear growth rates of crystals in supersaturated solutions and supercooled melts (single-component and binary) are summarized and compared with experimental data. This article is part of the theme issue 'Heterogeneous materials: metastable and non-ergodic internal structures'.

Heterogeneous materials: metastable and non-ergodic internal structures

This issue is concerned with structural and phase transitions in heterogeneous and composite materials, the effects of external magnetic fields on these phenomena and the macroscopic properties and behaviour of materials with isotropic and anisotropic internal structures. Using experimental, theoretical and computer methods, these transitions are studied at the atomic and mesoscopic levels. The fundamental specific feature of structural transitions in many heterogeneous media consists of the fact that these transitions are stacked for a long time in non-equilibrium states that appear due to either macroscopic dissipative processes (an alternating magnetic field or hydrodynamic flow, for instance) or system lifetime in a metastable state. It is important to explain and describe these transitional states using the general approach of non-equilibrium physical mechanics. The review and research articles in the issue will cover the whole spectrum of scales (from nano to macro) and materials (from metastable liquids to biological polymers) in order to exhibit recently developed trends in the field of heterogeneous materials. Atomistic modelling, structuring induced by external magnetic fields and hydrodynamic flows, metastable and non-ergodic states, mechanical properties and phenomena in heterogeneous materials-all these are covered. This article is part of the theme issue 'Heterogeneous materials: metastable and non-ergodic internal structures'.