Condensation Nucleation Research Articles

Nucleation landscape of biomolecular condensates.

All structures within living cells must form at the right time and place.This includes condensates such as the nucleolus, Cajal bodies and stress granules, which form via liquid-liquid phase separation of biomolecules, particularly proteins enriched in intrinsicallydisordered regions (IDRs)1,2. In non-living systems, the initial stages of nucleated phase separation arise when thermal fluctuations overcome an energy barrier due to surface tension. This phenomenon can be modelled by classical nucleation theory (CNT), which describes how the rate of droplet nucleation depends on the degree of supersaturation, whereas the location at which droplets appear is controlled by interfacial heterogeneities3,4. However, it remains unknown whether this framework applies in living cells, owing to the multicomponent and highly complex nature of the intracellular environment, including the presence of diverse IDRs, whose specificity of biomolecular interactions is unclear5-8. Here we show that despite this complexity, nucleation in living cells occurs through a physical process similar to that in inanimate materials, but the efficacy of nucleation sites can be tuned by their biomolecular features. By quantitatively characterizing the nucleation kinetics of endogenous and biomimetic condensates in living cells, we find that key features of condensate nucleation can be quantitatively understood through a CNT-like theoretical framework. Nucleation rates can be substantiallyenhanced by compatible biomolecular (IDR) seeds, and the kinetics of cellular processes can impact condensate nucleation rates and specificity of location. This quantitative framework sheds light on the intracellular nucleation landscape, and paves the way for engineering synthetic condensates precisely positioned in space and time.

On the Fokker–Planck approximation in the kinetic equation of multicomponent classical nucleation theory

We examine the validity of the Fokker–Planck equation with linear force coefficients as an approximation to the kinetic equation of nucleation in homogeneous isothermal multicomponent condensation. Starting with a discrete equation of balance governing the temporal evolution of the distribution function of an ensemble of multicomponent droplets and reducing it (by means of Taylor series expansions) to the differential form in the vicinity of the saddle point of the free energy surface, we have identified the parameters whereof the smallness is necessary for the resulting kinetic equation to have the form of the Fokker–Planck equation with linear (in droplet variables) force coefficients. The “non-smallness” of these parameters results either in the appearance of the third or higher order partial derivatives of the distribution function in the kinetic equation or in its force coefficients becoming non-linear functions of droplet variables, or both; this would render the conventional kinetic equation of multicomponent nucleation and its predictions inaccurate. As a numerical illustration, we carried out calculations for isothermal condensation in five binary systems of various non-ideality at T=293.15 K: 1-butanol–1-hexanol, water–methanol, water–ethanol, water–1-propanol, water–1-butanol. Our results suggest that under typical experimental conditions the kinetic equation of binary nucleation of classical nucleation theory may require a two-fold modification and, hence, the conventional expression for the steady-state binary nucleation rate may not be adequate for the consistent comparison of theoretical predictions with experimental data.

Numerical investigation on deposition rate of mechanically mixed ash particles in an entrained flow reactor

AbstractHomogeneous nucleation and heterogeneous condensation of salt vapors near the wall of superheater or reheater tubes greatly improve the sticky propensity of deposited surfaces or impacting particles and then lead to a fast‐growing rate of initial layer. Based on the deposition experiment of mechanically mixed KCl + SiO2 particles in an entrained flow reactor, a numerical model with considering the salt particle vaporization, salt vapor condensation, and its effect on SiO2 deposition is proposed for simulating impaction and adhesion behavior of molten and solid particles. The results show that deposits are mainly from the impaction of solid and molten particles and the condensation of salt vapors. The condensation rate of salt vapors in the stagnation zone at the windward side of metal probe is maximal. In comparison with the overall deposition rate, although the amount of salt vapors condensed on the probe surface can be negligible, it significantly affects the deposition rate of SiO2 particles. The coupled CFD model over a link between particle sticking and salt vapor condensation could well simulate the deposition behavior of mechanically mixed ash particles under rich‐salt‐vapor atmosphere.

Detection and experimental analysis of the molecular clusters distribution during wall condensation

Through Rayleigh scattering light intensity measurement, this paper investigates the transition state characteristics and microscopic evolution mechanism of the vapor water aggregate into liquid water in a narrow space near the subcooling wall. With the experimental analysis, clusters distribution layers with different characteristics in the several hundred microns width region near the condensation wall are found, especially the layer in which the average clusters size may maintain at a stable level. Since the same-sizes clusters layer, a relatively stable cluster diffusion region forms, and the size of the clusters diffusing from the region is evaluated to be about dozens of water molecules. The experimental analysis also shows that the cluster distribution and diffusion process in the near wall region are affected by condensation conditions and wall microstructure, which will further enrich the micro mechanism of the condensation nucleation process, even bring new methods to intensify the process.

A Study of Moist Air Condensation Characteristics in a Transonic Flow System

When water vapor in moist air reaches supersaturation in a transonic flow system, non-equilibrium condensation forms a large number of droplets which may adversely affect the operation of some thermal-hydraulic equipment. For a better understanding of this non-equilibrium condensing phenomenon, a numerical model is applied to analyze moist air condensation in a transonic flow system by using the theory of nucleation and droplet growth. The Benson model is adopted to correct the liquid-plane surface tension equation for realistic results. The results show that the distributions of pressure, temperature and Mach number in moist air are significantly different from those in dry air. The dry air model exaggerates the Mach number by 19% and reduces both the pressure and the temperature by 34% at the nozzle exit as compared with the moist air model. At a Laval nozzle, for example, the nucleation rate, droplet number and condensation rate increase significantly with increasing relative humidity. The results also reveal the fact that the number of condensate droplets increases rapidly when moist air reaches 60% relative humidity. These findings provide a fundamental approach to account for the effect of condensate droplet formation on moist gas in a transonic flow system.

Interactions of organosulfates with water vapor under sub- and supersaturated conditions

Abstract. Organosulfates (OSs) are important constituents of secondary organic aerosols, but their hygroscopic properties and cloud condensation nucleation (CCN) activities have not been well understood. In this work we employed three complementary techniques to characterize interactions of several OSs with water vapor under sub- and supersaturated conditions. A vapor sorption analyzer was used to measure mass changes in OS samples with relative humidity (RH, 0 %–90 %); among the 11 organosulfates examined, only sodium methyl sulfate (methyl-OS), sodium ethyl sulfate (ethyl-OS), sodium octyl sulfate (octyl-OS) and potassium hydroxyacetone sulfate were found to deliquesce as RH increased, and their mass growth factors at 90 % RH were determined to be 3.65 ± 0.06, 3.58 ± 0.02, 1.59 ± 0.01 and 2.20 ± 0.03. Hygroscopic growth of methyl-, ethyl- and octyl-OS aerosols was also studied using a humidity tandem differential mobility analyzer (H-TDMA); continuous hygroscopic growth was observed, and their growth factors at 90 % RH were determined to be 1.83 ± 0.03, 1.79 ± 0.02 and 1.21 ± 0.02. We further investigated CCN activities of methyl-, ethyl- and octyl-OS aerosols, and their single hygroscopicity parameters (κccn) were determined to be 0.459 ± 0.021, 0.397 ± 0.010 and 0.206 ± 0.008. For methyl- and ethyl-OS aerosols, κccn values agree reasonably well with those derived from H-TDMA measurements (κgf) with relative differences being < 25 %, whereas κccn was found to be ∼ 2.4 times larger than κgf for octyl-OS, likely due to both the solubility limit and surface tension reduction.

Advances of Silver Iodide Seeding Agents for Weather Modification

Silver iodide (AgI) is the most widely-used seeding agent in field experiment and operations of weather modification. There are several nucleation modes for AgI seeding agents, and the nucleation process is affected by many factors including atmospheric temperature, humidity, particle size, composition of seeding agent and its particle generation method. The nucleation efficiency, nucleation modes affect the number of nucleated crystals, and thus affect the seeding effect.Through laboratory research of cloud chamber and theoretical calculation, the critical embryo sizes required for different nucleation mechanisms, nucleation threshold temperature, nucleation mechanisms, factors effecting nucleation for AgI and with some other added materials are analyzed. It is currently accepted that there are four nucleation mechanisms, including deposition, condensation freezing, contact freezing and immersion freezing nucleation. And it is recognized that nucleation processes depend on the varied temperature, humidity, and cloud conditions that can be encountered in the atmosphere. Immersion nucleation refers to nucleation of freezing by a particle immersed in water, and deposition nucleation refers to nucleation of the ice phase from the vapor, contact nucleation refers to the nucleation of the freezing induced by a particle during first contact with supercooled water, and condensation freezing is the nucleation of the freezing from the condensation of vapor to liquid droplet.The ice-nucleating properties and nucleation effectiveness of cloud seeding materials produced by burning acetone solutions or pyrotechnics of AgI and other materials are tested in a wind-tunnel cloud chamber test facility along with isothermal cloud chamber and dynamic cloud chamber. Through laboratory test, different formulations are compared, and high nucleation efficiency of seeding materials are selected, and nucleation features for four nucleation modes separately are obtained.Silver iodide seeding cloud model are based on a combination of theory and laboratory results. The ice nucleation schemes employed in cloud models vary widely. Hsie's seeding scheme simulated both contact-freezing and deposition nucleation on the laboratory measured effectiveness spectra of nucleus. Meyers seeding scheme considers all the four processes on cloud chamber results of ice-forming processes by AgI. In China, AgI seeding models are developed since the 1990s, either similar with Hsie's seeding scheme on three nucleation mechanisms, or similar with Meyers seeding scheme on four nucleation mechanisms.

Numerical simulation on the growth of polydisperse fine SiO2 particles by water vapor condensation

The three-stage model was utilized to analyze the law of grown particle size distribution for amplification of polydisperse SiO2 particles in actual water vapor environment. The predicted results demonstrate that the dispersity is narrowed compared with experimental data and the difference between them is large at higher level of supersaturation. Moreover, representative affecting factors on particle growth are discussed. Specifically, the increase of particle number concentration is the significantly negative effect on the particle growth. It leads to the failure of nucleation of small size particles and condensation on large size particles resulting in the several times reduction of arithmetic mean droplet size. Additionally, the improvement of the peak of initial particle size and inlet gas flow temperature on the condition of heating gas flow is not beneficial to particle growth. Besides, the reduction of surface tension of liquid phase can improve the growth strongly at lower level of supersaturation.

Single-molecule and ensemble methods to probe RNP nucleation and condensate properties.

Biomolecular condensates often consist of intrinsically disordered protein and RNA molecules, which together promote the formation of membraneless organelles in cells. The nucleation, condensation, and maturation of condensates is a critical yet poorly understood process. Here, we present single-molecule and accompanying ensemble methods to quantify these processes more comprehensively. In particular, we focus on how to properly design and execute a single-molecule nucleation assay, in which we detect signals arising from individual units of fluorescently labeled RNA-binding proteins associating with an RNA substrate. The analysis of this data allows one to determine the kinetics involved with each step of nucleation. Complemented with meso-scale techniques that measure the biophysical properties of ribonucleoprotein condensates, the methods described herein are powerful tools that can be adopted for studying any protein-RNA interactions that promote phase separation.

Valency and Binding Affinity Variations Can Regulate the Multilayered Organization of Protein Condensates with Many Components.

Biomolecular condensates, which assemble via the process of liquid–liquid phase separation (LLPS), are multicomponent compartments found ubiquitously inside cells. Experiments and simulations have shown that biomolecular condensates with many components can exhibit multilayered organizations. Using a minimal coarse-grained model for interacting multivalent proteins, we investigate the thermodynamic parameters governing the formation of multilayered condensates through changes in protein valency and binding affinity. We focus on multicomponent condensates formed by scaffold proteins (high-valency proteins that can phase separate on their own via homotypic interactions) and clients (proteins recruited to condensates via heterotypic scaffold–client interactions). We demonstrate that higher valency species are sequestered to the center of the multicomponent condensates, while lower valency proteins cluster towards the condensate interface. Such multilayered condensate architecture maximizes the density of LLPS-stabilizing molecular interactions, while simultaneously reducing the surface tension of the condensates. In addition, multilayered condensates exhibit rapid exchanges of low valency proteins in and out, while keeping higher valency proteins—the key biomolecules involved in condensate nucleation—mostly within. We also demonstrate how modulating the binding affinities among the different proteins in a multicomponent condensate can significantly transform its multilayered structure, and even trigger fission of a condensate into multiple droplets with different compositions.

Preferential Vapor Nucleation on Hierarchical Tapered Nanowire Bunches.

Controlling vapor nucleation on micro-/nanostructured surfaces is critical to achieving exciting droplet dynamics and condensation enhancement. However, the underlying mechanism of nucleation phenomena remains unclear because of its nature of nanoscale and transience, especially for the complex-structured surfaces. Manipulating vapor nucleation via the rational surface design of micro-/nanostructures is extremely challenging. Here, we fabricate hierarchical surfaces comprising tapered nanowire bunches and crisscross microgrooves. Nanosteps are formed around the top of the nanowire bunches, where the nanowires all around agglomerate densely because of surface tension. The theoretical analysis and molecular dynamics simulation show that nanostep morphologies that are around the top of the nanowire bunches can enable a lower energy barrier and a higher nucleation capability than those of the sparsely packed nanowires at the center and bottom of the nanowire bunches. Vapor condensation experiments demonstrate that the nucleation preferentially occurs around the top of the nanowire bunches. The results provide guidelines to design micro-/nanostructures for promoting vapor nucleation and droplet removal in condensation.

Lattice Boltzmann simulation of dropwise condensation on the microstructured surfaces with different wettability and morphologies

Dropwise condensation heat transfer on the different microstructured surfaces is simulated by 2D hybrid thermal lattice Boltzmann method. Dynamic behaviors of condensate droplets on the microstructured surfaces are investigated. Moreover, the influence of surface wettability, surface microstructure and subcooling degree on the condensate dynamics and heat transfer performance is analyzed. Our numerical results demonstrate that condensate microdroplets preferentially appear to form at the valley or side of micropillar arrays. As the condensation progresses, the condensate mass and condensation rate of the superhydrophobic surface eventually exceed those of the hydrophilic and hydrophobic surfaces. The heat flux of superhydrophobic surface with θa = 150.7° is the highest. The local heat flux at the top of micropillars is much higher than that at the bottom of micropillars. The microstructured surface with a smaller spacing is beneficial to increasing the condensate nucleation population, whereas a larger spacing can strengthen the jumping abilities of coalesced droplets. During condensation, the average heat flux fluctuates locally with time caused by the merging and jumping dynamics. Although the superhydrophobic surface with triangle microstructure improves the jumping height of coalesced droplets, the heat flux of triangle microstructured surface is significantly smaller over that of the square and semicircular microstructured surfaces. Despite a greater subcooling degree facilitates the condensation heat transfer, it has a negative influence on the jumping height of droplets. This numerical study illustrates the microscopic mechanism of dropwise condensation heat transfer in detail.

Nucleated transcriptional condensates amplify gene expression.

Membraneless organelles or condensates form through liquid-liquid phase separation1-4, which is thought to underlie gene transcription through condensation of the large-scale nucleolus5-7 or in smaller assemblies known as transcriptional condensates8-11. Transcriptional condensates have been hypothesized to phase separate at particular genomic loci and locally promote the biomolecular interactions underlying gene expression. However, there have been few quantitative biophysical tests of this model in living cells, and phase separation has not yet been directly linked with dynamic transcriptional outputs12,13. Here, we apply an optogenetic approach to show that FET-family transcriptional regulators exhibit a strong tendency to phase separate within living cells, a process that can drive localized RNA transcription. We find that TAF15 has a unique charge distribution among the FET family members that enhances its interactions with the C-terminal domain of RNA polymerase II. Nascent C-terminal domain clusters at primed genomic loci lower the energetic barrier for nucleation of TAF15 condensates, which in turn further recruit RNA polymerase II to drive transcriptional output. These results suggest that positive feedback between interacting transcriptional components drives localized phase separation to amplify gene expression.

Heterogeneous nucleation of argon vapor on the nanostructure surface with molecular dynamics simulation

The study of vapor condensation on surface has important engineering significance. The condensation nucleation process of vapor on substrates with different pillar structures is studied through molecular dynamic simulation. The condensation droplets on the pillar with various heights and solid fractions exhibit Wenzel state, Cassie state and the transformation from Wenzel state to Cassie state. The results show that the condensation efficiencies are correlated with the state of droplet and it is explained from the perspective of heat transfer. For the Wenzel state, the droplet fills the gap of pillar and the form of the heat conduction change with the growth of cluster. In the initial of condensation droplets, the heat conduction is similar with various heights of pillar. As the condensation droplets grow, the efficiency of heat conduction enhances with the increasing of height of pillar. For the Cassie state, the form of heat conduction is perpetual during the condensation process with the thermal resistance of the droplet dominated due to the droplets suspended on the pillar. The efficiency of heat transfer is insensitive to the height of pillar. The form of heat conduction for the transformation state transforms from Wenzel state to Cassie state leading to the reduction of condensation rate. The droplet formed in the Wenzel state has the higher transfer efficiency than the Cassie one.

Numerical study on two-phase expansion performance and quantitative analysis of wetness loss in cryogenic turbo-expander

Cryogenic two-phase turbo-expander becomes a prevalent option in process design of high efficiency cryogenic liquefaction system, especially in modern air separation system and helium system. Computational code of turbine off-design performance prediction in overheated expansion and non-equilibrium condensation module are well established in our former works, and the mechanism of nucleation and droplet growth in spontaneous condensation is revealed. However, the analysis between single and two-phase expansion as well as wetness influence on turbine performance is not conducted yet. In this study, we investigate a turbo-expander performance analysis in a wide temperature range (Tin = 118 K ~ 98 K), and present the wetness loss assessment and quantitative calculation in two-phase expansion process. By referring to the previous off-design performance of overheated expansion, the two-phase expansion performance under different pressure ratio and rotational speed is obtained and compared. According to the continuous inlet temperature drop process, the expansion is categorized into three types and their zone boundaries are divided. The transitional feature of unexpected change is frequently figured out on the expansion zone boundary, and the reaction ratio and pressure ratio in diffuser during the continuous inlet temperature drop process are analyzed detailed, which shows the unique and significant role of wetness effect in two-phase expansion. Based on the experimental wetness loss assessment method established by Baumann in wet steam turbine, two numerical assessments by using overheated expansion and equilibrium condensation model as reference expansion are analyzed. To quantitatively analyze the wetness loss influence, the efficiency drop caused by wetness and the relative wetness loss are calculated by eliminating other component losses in turbine, and the relative wetness loss is fitted with outlet liquid mass fraction mathematically to assist future design of cryogenic two-phase turbo-expanders.

Confined Growth and Controlled Coalescence/Self-Removal of Condensate Microdrops on a Spatially Heterogeneously Patterned Superhydrophilic-Superhydrophobic Surface.

Manipulating condensate nucleation, growth, coalescence, and self-removal via bionic super-wettability surfaces has attracted intensive interest because of their significance in fundamental research and technological innovations, for example, water harvesting, power generation, air conditioning, and thermal management. However, it is still a challenge to simultaneously realize confined growth, coalescence, and self-ejection of condensate microdrops, which has not been reported to date. Here, we propose and demonstrate a type of new and more efficient coalescence/self-removal method based on spatially confined growth/coalescence/self-ejection of condensate microdrops, which can be realized using a rationally designed superhydrophobic surface with spatially heterogeneously patterned superhydrophilic microdots (SMDs). Exemplified by superhydrophobic closely packed zinc oxide nanoneedles with SMD patterns, we investigate how the geometric parameters of SMD patterns be designed to simultaneously realize the spatially confined growth/coalescence/self-ejection of patterned microdrops, which are rationalized via theoretical analyses.

Improved modeling of heat transfer in dropwise condensation

Dropwise condensation has drawn significant attention due to its efficient heat transfer performance compared to filmwise condensation. In this paper, typical experimental data for dropwise condensation on smooth hydrophobic surfaces were collected as well as the typical available heat transfer models. The comparisons between the prediction results and the experimental data indicated that the existing models were not generally applicable to various conditions. A new model for a vertical smooth surface was developed to predict the heat transfer characteristics of dropwise condensation. The new model was based on the nucleation condensation mechanism, and the total heat transfer on the surface includes the heat through all the droplets and the heat through the surface between the droplets. For the latent heat through the droplets the effect of the contact angle was taken into consideration on the basis of the nucleation condensation mechanism. The surface area between the droplets on the surface was thought to be the bare surface, and sensible heat transferred on the bare surface and the droplets surface was viewed as forced convection heat transfer. The calculation results from the model show that, although the heat transferred by forced convection is greatly dependent on the experimental parameters, it is three orders of magnitude smaller than the latent heat through the droplets. Comparisons show that the present model has better prediction precision, with an error range of -35–20% for 87.39% of the data and an error range of -35–25% for 90.37% of the data. The findings obtained from the model suggest that the heat transfer rate and the critical nucleation radius for a single droplet and the droplet size distribution are remarkably affected by the contact angle. In fact, a smaller contact angle enhances the condensation heat transfer and increases the nucleation density. In addition, the thickness of the promoter layer weakens the condensation heat transfer and decreases the nucleation density.

Modeling and assessment of two-phase transonic steam flow with condensation through the convergent-divergent nozzle

In the Fukushima accident, it was found that the Reactor Core Isolation Cooling (RCIC) system played a crucial role in delaying core meltdown by almost three days in Fukushima Daiichi Unit 2, because of self-regulated operation of the steam driven RCIC turbine-pump injection system. Steam flow in the convergent-divergent nozzles of the RCIC Terry turbine is two-phase non-equilibrium transonic flow with homogenous nucleation condensation. To more accurately predict the dynamic process and behavior of the transonic compressible steam flow, a one-dimensional transient two-phase analytical model is presented. A simplified four-fluid model was employed in the present work with the consideration of four separate fluid fields: vapor, liquid film, entrained droplets and condensed droplets. The mass, momentum and energy interactions between the fluids were considered and modeled. An extended seven-equation non-equilibrium critical flow model was developed to obtain the critical pressure and velocities of each phase at the nozzle throat. To predict the wetness in the divergent section, a mechanistic nucleation condensation model was integrated in the nozzle analysis model, considering the generation and consequent growth of droplets. The governing differential equations on a staggered grid were discretized using the second-order Lax-Wendroff scheme with a flux limiter, and the Semi-Implicit Method for Pressure-Linked Equation (SIMPLE) algorithm was employed to solve the discrete linear system. To demonstrate the predictability and reliability of the physical models and the numerical method proposed in the present work, three representative nozzles were modeled and simulated. The results show good agreement with the available experimental data, even for condensation shock.

Therapeutics-how to treat phase separation-associated diseases.

Liquid–liquid phase separation has drawn attention as many neurodegeneration or cancer-associated proteins are able to form liquid membraneless compartments (condensates) by liquid–liquid phase separation. Furthermore, there is rapidly growing evidence that disease-associated mutation or post-translational modification of these proteins causes aberrant location, composition or physical properties of the condensates. It is ambiguous whether aberrant condensates are always causative in disease mechanisms, however they are likely promising potential targets for therapeutics. The conceptual framework of liquid–liquid phase separation provides opportunities for novel therapeutic approaches. This review summarises how the extensive recent advances in understanding control of nucleation, growth and composition of condensates by protein post-translational modification has revealed many possibilities for intervention by conventional small molecule enzyme inhibitors. This includes the first proof-of-concept examples. However, understanding membraneless organelle formation as a physical chemistry process also highlights possible physicochemical mechanisms of intervention. There is huge demand for innovation in drug development, especially for challenging diseases of old age including neurodegeneration and cancer. The conceptual framework of liquid–liquid phase separation provides a new paradigm for thinking about modulating protein function and is very different from enzyme lock-and-key or structured binding site concepts and presents new opportunities for innovation.

An Early Association between the α-Helix of the TEAD Binding Domain of YAP and TEAD Drives the Formation of the YAP:TEAD Complex.

The Hippo pathway is an evolutionarily conserved signaling pathway that is involved in the control of organ size and development. The TEAD transcription factors are the most downstream elements of the Hippo pathway, and their transcriptional activity is regulated via the interaction with different co-regulators such as YAP. The structure of the YAP:TEAD complex shows that YAP binds to TEAD via two distinct secondary structure elements, an α-helix and an Ω-loop, and site-directed mutagenesis experiments revealed that the Ω-loop is the "hot spot" of this interaction. While much is known about how YAP and TEAD interact with each other, little is known about the mechanism leading to the formation of a complex between these two proteins. Here we combine site-directed mutagenesis with pre-steady-state kinetic measurements to show that the association between these proteins follows an apparent one-step binding mechanism. Furthermore, linear free energy relationships and a Φ analysis suggest that binding-induced folding of the YAP α-helix to TEAD occurs independently of and before formation of the Ω-loop interface. Thus, the binding-induced folding of YAP appears not to conform to the concomitant formation of tertiary structure (nucleation-condensation) usually observed for coupled binding and folding reactions. Our findings demonstrate how a mechanism reminiscent of the classical framework (diffusion-collision) mechanism of protein folding may operate in disorder-to-order transitions involving intrinsically disordered proteins.