Physical effects of crowdant size and concentration on collective microtubule polymerization

Abstract

The polymerization of cytoskeletal filaments is regulated by both biochemical pathways, as well as physical factors such as crowding. The effect of crowding in vivo emerges from the density of intracellular components. Due to the complexity of the intracellular environment, most studies are based on either in vitro reconstitution or theory. Crowding agent (crowdants) size has been shown to influence polymerization of both actin and microtubules (MTs). Previously, the elongation rates of MT dynamics observed at single filament scale were reported to decrease with increasing concentrations of small but not large crowdants, and this correlated with in vivo viscosity increases. However, the exact nature of the connection between viscosity, crowdant size, nucleation, and MT elongation has remained unclear. Here, we use in vitro reconstitution of bulk MT polymerization kinetics and microscopy to examine the collective effect of crowdant molecular weight, volume occupancy, and viscosity on elongation and spontaneous polymerization. We find MT elongation rates obtained from bulk polymerization decrease in the presence of multiple low-molecular weight (LMW) crowdants, while increasing with high-molecular weight (HMW) crowdants. Lattice Monte Carlo simulations of an effective model of collective polymerization demonstrate reduced polymerization rates arise due to decrease in monomer diffusion due to small-sized crowdants. However, MT polymerization in the absence of nucleators, de novo, shows a crowdant size independence of polymerization rate and critical concentration, depending solely on concentration of the crowdant. In microscopy, we find LMW crowdants result in short but many filaments, while HMW crowdants increase filament density, but have little effect on lengths. The effect of crowdant volume fraction ${\varphi }_C$ and size in de novo polymerization match simulations, demonstrating crowdants affect elongation independent of nucleation. Thus, the effect of viscosity on collective MT dynamics, i.e., filament numbers and lengths, shows crowdant size dependence for elongation, but independence for de novo polymerization.

Significance

Previous work has demonstrated that crowdants altered single microtubule (MT) filament elongation rates in vitro based on small-sized decreases, while large increases were seen to be correlated to increased viscosity in vivo. Here, we examine MT polymerization dynamics with crowdants of different sizes by in vitro reconstitution and relate crowdant size, concentration, and viscosity to MT elongation and nucleation. While polymerization rate in the presence of nucleating “seeds” changes depending on crowdant molecular weight (size), de novo polymerization without seeds increases regardless of crowdant size. Using a combination of quantitative microscopy and simulations, we find this crowdant size independence emerges from the collective effect of rate limited MT nucleation and diffusion-limited elongation influenced by crowdant size.

Introduction

Microtubules (MTs) are intracellular cytoskeletal filaments vital for cell physiology, growth, and division. The filaments polymerize by the assembly of GTP bound heterodimers of $\alpha$- and $\beta$-tubulin. The polymerization dynamics of MTs are described by nucleation-dependent polymerization (NDP) kinetics involving an initial nucleation stage followed by elongation (1,2,3). The concentration of subunits above the critical concentration (c^∗) leads to the formation of stable nuclei, while elongation occurs by the addition of subunits to the ends accompanied by GTP hydrolysis (4). At single-filament level, MTs undergo “dynamic instability,” defined as the transition between growing and shrinking states (5). The in vivo growth of MTs is expected to deviate from such ideal conditions in the cytoplasm due to both biochemical and physical factors. Biochemical factors such as centrosomes, microtubule-associated proteins, and accessory proteins have been extensively shown to regulate both nucleation and polymerization (6,7,8,9). Physical theories of polymerization and the effect of solutes and colloids were initially based on the Asakura Oosawa model that modeled the effect of self-aggregation of macromolecules above a critical concentration (10,11). Theoretically, the change in volume fraction due to crowding was shown to affect equilibrium polymer distribution of simplified rod-like polymers by extending the Asakura Oosawa model (12). In experiments, the elongation rates of actin filament in the presence of seeds was seen to reduce with low-molecular weight (LMW) crowdants, while they were increased by high-molecular weight (HMW) crowdants (13). In contrast, the rate of de novo bulk polymerization of actin in the absence of nucleating seeds was unaffected by LMW crowdants, while HMW crowdants increased rates (14). A comparable study on single filament MT polymerization dynamics in vitro has demonstrated that LMW crowdants resulted in a decrease of elongation rates, while HMW crowdants increased them, in a concentration-dependent manner (15). An in vivo study demonstrated osmotic stress induced increases in cytoplasmic concentration resulted in a reduction in MT growth rates in Schizosaccharomyces pombe cells, and it correlated with increased viscosity, comparable with the in vitro effects of glycerol on MT elongation (16). While a size dependence of crowdants on cytoskeletal polymerization dynamics has been observed for actin and microtubules, an explanation of how crowdant concentration, viscosity, and available volume combine to influence MT polymerization remains unclear.

Crowding is ubiquitous and a physical effect due to a reduction in the available volume for movement, originating from the concentration of molecules, macromolecular assemblies, and organelles (17,18,19). The specific effects of crowding effects are varied ranging from excluded volume, depletion effect, and viscosity changes (20). Of these, cytoplasmic viscosity $\left({\eta }_{cyto}\right)$ has been measured in many cellular systems with wide ranging values reported, from 100 to 1000 times greater than water ($\sim$1 cP). The cytoplasmic viscosity of multiple model cells has been reported, with newt cell cytoplasm at 280 cP (21), embryos of Caenorhabditis elegans in the one-cell stage at ${10}^2$ to ${10}^3$ cP (22,23,24) and $\sim$10 cP for Escherichia coli cytosol (25,26). A distinct probe size dependence of the measure itself was also reported due to the structured nature of the cytoplasm (26,27). At the nanometer scale, cytoplasmic viscosity approaches $\sim$1 cP, that of water (28). To disambiguate the effects of the cytoplasmic environment, in vitro reconstitution approaches have been widely used (19). Most studies have considered the effects of cosolute molecules that are either of the same size or larger than the “test” molecule under investigation, but otherwise inert, with the effect then described as macromolecular crowding (29,30). Cosolutes with molecular weights smaller than the test molecule with correspondingly smaller hydrodynamic radii have also, however, been shown to act as crowding agents (31). Small crowdants are reported to act differently from large crowdants, as seen in the decrease in protein-protein association rates with increasing concentrations of ethylene glycol and PEG 200 (32). Glycerol, much smaller in size to the tracer macromolecules—proteins and DNA—influenced both diffusive mobility and conformation (33). Ethylene glycol was used as a small crowdant to examine self-assembly and oligomer formation (34). Based on these contrasting roles, an MW threshold of 1000 Da was proposed to classify crowdants into LMW and HMW crowdants (35). Actin polymerization in the presence of a small-molecule cosolute such as glycerol was shown to result in decreased polymerization, despite the increase in viscosity (14). Based on these studies, we proceeded to consider those molecules to be crowdants that are: 1) highly water soluble, 2) inert with respect to the bio-molecular process being studied (e.g., cytoskeletal polymerization), and 3) able to reproduce some aspect of the crowded cytoplasm, e.g., density of packing. Based on the 1 kDa threshold described previously (35), we classify the molecules tested into LWM and HMW crowdants.

The reduced volume, arising from the presence of crowdants, is naturally expected to influence intermolecular interaction dynamics, since most biochemical reactions rely on diffusion for transport (36). Measurements of protein diffusion in vitro have demonstrated a decrease in mobility with tracer size in a crowded environment (37). This effect was modeled by hard-shell interactions of crowdants and tracers based on previously developed scaled particle theory (38,39). The diffusion of globular proteins was found in simulations to be altered if the filaments were random or aligned, to mimic the effect of cytoskeletal filaments (40). Simulations predicted a dramatic decrease in protein association rate in the presence of small- but not large-sized crowdants (32,41). Experimental tests of size dependence show that, while amyloid fiber association rates were enhanced by small crowdants in a concentration-dependent manner (42), a reduction in elongation rates was observed for polymerizing microtubules (16) and actin (14). These divergent experimental observations could be better explained by simulations that unite concepts from polymerization dynamics and diffusion in crowded environments.

Simulations using lattice-based approaches with obstacles have been used to examine the effect of mobility of such crowdants (43,44) and combined with aggregation of the obstacles (45). Increasing the packing fraction of the obstacles or crowdants, below percolation threshold, was observed to result in anomalous diffusion, when obstacles were of the same size as the tracer (46). Crowdant mobility and size both influence diffusivity with the anomalous diffusion exponent affected more strongly by small compared with large in a 3D lattice simulation (47). Combining the processes of diffusion and association, a computational and theoretical study demonstrated that, when association rates were high but encounter low, i.e., diffusion limited, crowdant volume fraction increases, resulting in a decrease in association rates, while effective reaction rates were increased by packing when they were inherently rate limited (48). A mixture of small and large crowdants in simulations to mimic cytoplasmic conditions demonstrated that association rates increased at low crowdant volume fractions $\left({\varphi }_c\right)$ and were reduced at high volume fractions (49). Indeed, we have shown in simulations that crowdant structure formation can also alter association dynamics in lattice simulations at the same volume fraction (50). Thus, from simulations we expect crowdant size to play a role in cytoskeletal filament polymerization due to the nature of monomer diffusion, nucleation, and end-growth.

Here, we have examined the physical effects of crowdant size and concentration on MT polymerization dynamics by in vitro reconstitution of bulk and microscopic polymerization. We show that elongation rates of MT filaments from nucleating seeds are decreased by LMW crowdants, and increased by HMW crowdants in turbidimetry. Simulations predict the effect of small crowdant packing in terms of reduced MT lengths, comparable with reduced elongation rates, with decreased diffusivity of subunits. Large crowdants are predicted to affect diffusion to a lesser degree. Experiments of MT polymerization in the absence of seeds, i.e., de novo, is increased by crowdant volume fraction, independent of the size of the crowdant molecule. By measuring the effect of crowdants on the critical concentration from bulk turbidimetry and filament lengths and numbers from microscopy, we examine how crowdant concentration, size and the solution viscosity affect MT elongation and nucleation. We find that aspects of our simulation can explain the results from quantitative microscopy and compare the effects on lengths, polymerization rates in terms of packing, crowdant size, and viscosity.

Material and methods

Purification and polymerization kinetics of brain tubulin with crowdants

Tubulin was purified from freshly sacrificed goat brains as described previously (51), based on a temperature-based polymerization depolymerization cycling method (52). All fine chemicals were sourced from Sigma-Aldrich (St. Louis, MO, USA), unless otherwise stated.

Purified brain tubulin polymerization kinetics were measured by measuring absorbance at 340 nm using 96-well half-area UV-transmissible flat bottomed plates (Corning, NY, USA) in a microtiter plate reader (Varioskan Flash, Thermo Scientific, Waltham, MA, USA) with 20–60 s time interval typically over 20–40 min. Two kinds of experiments were performed: 1) MT elongation kinetics and 2) de novo polymerization kinetics in the presence of a range of crowdants with molecular weights spanning $\sim {10}^1$ to ${10}^5$ Da. The crowdants used were: ethylene glycol of MW of 62.07 Da, trimethylamine oxide (TMAO) MW of 75 Da, glycerol MW of 92 Da, polyethylene glycol (PEG) 400 MW of 400 Da, PEG 4000 MW of 3.5–4.5 kDa, Ficoll 400 MW of 400 kDa, PEG 20000 MW of 20 kDa, and bovine serum albumin (BSA) MW of 66.43 kDa. The crowdant MWs and sizes in terms of hydrodynamic radii demonstrate that increasing MW correlates with an increase in molecular size (Table 1).

Table 1.

Crowdant molecular weights and sizes

Crowdant	Molecular weight (Da)	Hydrodynamic radius $\left({\mathrm{r}}_{\mathrm{h}}\right)$ (nm)	Reference
LMW, small crowdants
Ethylene glycol	62	0.3	Vital de Oliveira and Gomide Freitas (95)
TMAO	75	0.264	Comez et al. (96)
Glycerol	92	0.31	Elamin and Swenson (97)
PEG 400	400	0.68	Scherrer and Gerhardt (98)
HMW, large crowdants
PEG 4000	4 × ${10}^3$	1.92	Sabirov et al. (99)
PEG 20000	2 × ${10}^4$	4	Devanand and Selser, Gokarn et al. (100,101)
BSA	6.6 × ${10}^4$	3.8	Yu et al. (102)
Ficoll 400	4 × ${10}^5$	7.26	Ranganathan et al. (94)

Hydrodynamic radius of crowdant molecules is correlated with their molecular weights based on the literature.

MT elongation kinetics

Bulk turbidimetry-based observation of MT elongation were optimized from a number of previous reports (53,54,55). Stable MT nucleation seeds were prepared by incubating 45 $\mu$M tubulin with 0.8 mM GMPCPP (NU-405L; Jena Bioscience, Jena, Germany) in BRB-80 buffer (80 mM PIPES, 1 mM EGTA, and 1 mM ${\text{MgCl}}_2$) at 37°C for 30 to 40 min, until the absorbance at 340 nm remained unchanged. The assembled MTs were then sheared by ultrasonication (Ultrasonic Cleaner Sonicator, Spire Automation & Innovation, Pimpri-Chinchwad, India) for 1 min. The seeds were further sheared by pipetting using 10 $\mu$L tips and their presence confirmed in interference reflection microscopy (IRM). The final MT seed solution was diluted 1:3, and 10 $\mu$L of the diluted seed solution was mixed with 20 $\mu$L of reaction mix (8 $\mu$M tubulin in 1 mM GTP, 5 mM ${\text{MgCl}}_2$, and BRB-80 along with a crowdant) and absorbance at 340 nm monitored at 37°C for 40 min at an interval of 60 s. The effect of crowdants on 8 $\mu$M tubulin polymerization was measured both in the absence of seeds, (−)seed, as blank (Fig. S1, dashed lines), and in their presence, (+)seed (Fig. S1, dotted lines). The difference between the (+) and (−)seeds was used to quantify elongation, $A_{elong}=A_{(+)seed}-A_{(-)seed}$. The concentration of tubulin was optimized such that it does not show polymerization in the absence of crowdants (Fig. S2), and below the critical concentration estimated from (−)seed polymerization dynamics of tubulin without crowdants (Fig. S3).

De novo polymerization kinetics

A 30 $\mu$L reaction mix was prepared containing 30 $\mu$M tubulin, 1 mM GTP, 5 mM ${\text{MgCl}}_2$, BRB-80, and crowdants, and incubated at 37°C. Blank samples contained everything except tubulin. Absorbance at 340 nm (${\mathrm{A}}_{340nm}$) was monitored at an interval of 20 s or 25 min.

Analysis and fitting of polymerization kinetics data

The ${\mathrm{A}}_{340nm}$ values were pathlength corrected to 1 cm and blank (polymerization solution only lacking tubulin) subtracted. Data from individual experiments were averaged over two independent replicates each with three technical replicates and mean absorbance (${\mathrm{A}}_t$) with standard deviation calculated and plotted as a function of time. The absorbance kinetics data was fit to a standard four-parameter model used previously to fit for nucleation-limited polymerization kinetics (51,56,57,58) as follows:

$$A_t=A_o+\frac{A_{\max }-A_o}{1+e^{-r·\left[\left(t-t_{1/2}\right)\right]}}$$ (1)

where r is the polymerization rate (${\min }^{-1}$), $t_{1/2}$ is the time at which absorbance is half-maximal (min), and $A_{\max }$ and $A_0$ are the maximal saturation and initial absorbance values, respectively (arbitrary units).

The normalized steady-state absorbance was calculated as the difference between the fit obtained maximal and minimal absorbance, $A_n=\left[A_{\max }-A_0\right]$, to correct for the initial values that differ between experiments. This was used as a parameter to calculate the total polymerized tubulin.

We used the polymerization rate (r) to determine critical concentration of tubulin, using an approach reported in previous literature (51,59,60). Polymerization rate and concentration of tubulin was fit to a straight line as follows:

$$r=k·C+\alpha$$ (2)

where C is the concentration of tubulin with k and $\alpha$ as the slope and y-intercept, respectively. The critical concentration (c^∗) then is the x-intercept of this line, i.e., $c\ast =-\alpha /k$. The normalized steady-state absorbance, ($A_n$), as a measure of total polymer mass was also used in a similar manner to determine the critical concentration, where, $A_n$ was used instead of r in Eq. 2.

The volume fraction $\left({\varphi }_c\right)$ in experiment for the crowdants of different molecular weights is based on the fractional volume approach taken in literature (48,61), and was determined using the following equation:

$${\varphi }_c=\frac{N_c·v_{molec}}{V_{tot}}$$ (3)

where ${\mathrm{V}}_{tot}$ is the total reaction volume, in this case 30 $\mu$L or 3 × ${10}^{19}$ ${\text{nm}}^3$, $v_{mol}$ is the volume occupied by an individual crowdant molecule, and $N_c$ is the number of crowdant molecules. By approximating all crowdants as spheres and assuming hard-shell interactions, i.e., spheres are impenetrable, $v_{mol}$ is calculated from the hydrodynamic radius ($r_h$, in nm) as:

$$v_{mol}=\frac{4}{3}\pi r_h^3$$ (4)

The concentrations of crowdants (C, g/100 mL) was used to calculate the number of molecules of the crowdant $N_c$ from the given volume, v = 30 $\mu$L, the molecular weight of the crowdant (MW, in g) and Avogadro’s number (${\mathrm{N}}_A$) as follows:

$$N_c=N_A\times \frac{3\times {10}^{-4}\times C}{MW}$$ (5)

Thus, the total volume of a given crowdant is simply the product $N_c\times V_c$.

The rate of MT elongation estimated from seed nucleated polymerization kinetics were fit using either of two functions depending on whether the rate increased or decreased: 1) a saturation model based loosely on Michaelis-Menten kinetics for rates that increased with viscosity:

$$r=\frac{r_{\max }\ast \eta }{{\eta }_{1/2}+\eta }$$ (6)

where $r_{\max }$ is the maximal rate and ${\eta }_{1/2}$ is the viscosity where the rate is half-maximal, and 2) a function is inversely proportional to viscosity, for decreasing rates:

$$r=c_1/\eta$$ (7)

where $c_1$ is a fitting constant. De novo polymerization rates were more heterogeneous due to the combination of the underlying multistep nucleation and elongation, and were fit to straight lines. All data analysis was done using Python (ver. 3.9.12) with Numpy (ver. 1.21.5) and Scipy (ver. 1.7.3) libraries.

Bead diffusion microrheology to estimate viscosity of crowdant solutions

Viscosity of different crowdant concentrations were estimated by a microrheology, using an approach we have previously described elsewhere (62) that we briefly summarize here. Polystyrene beads of 1 $\mu$m diameter (Polysciences, Warrington, PA, USA) were diluted in BRB-80 buffer in combination with a range of concentrations of crowdants and mounted in double-backed tape chambers with a circular hole sandwiched between a slide and coverslip. Time-series images were acquired using a 40× (NA 0.65 ELWD) lens using a Nikon Ti-E inverted microscopy (Nikon, Tokyo, Japan) coupled to a CCD camera Andor Clara2 (Andor Technology, Belfast, UK) every 0.5 s for 1 min (Fig. S4 A). The diffusion coefficient was estimated from fitting the plot of the mean-square displacement (MSD) with time (Fig. S4 B) using a MATLAB (Mathworks Inc., Natick, MA, USA) program DICOT https://github.com/CyCelsLab/DICOT (62), and the viscosity then estimated by substitution in the Stokes-Einstein relation $D=k_{B}T/6\pi \eta R$.

IRM of MTs

Tubulin polymerization was visualized in microscopic round chamber setups as shown (Fig. S5), created using cleaned microscope slide and no. 1.5 coverslips with a thickness of 0.17 mm (VWR, Avantor, Radnor, PA, USA). The coverslips were pretreated with 1 N HCl at 50°C for 5 h, followed by rigorous washing with double distilled water to remove any traces of acid, followed by 70% ethanol wash. Tubulin (20 $\mu$M) in the presence of 1 mM GTP, 5 mM ${\text{MgCl}}_2$, and different crowdant concentrations in BRB-80 buffer was added into the round chambers created by double backed tape on slides, and were sandwiched by the coverslips.

Samples are imaged using a 100× (NA 1.4, oil) lens on a Nikon Eclipse Ti-E inverted microscope at 37°C by a temperature control system (Okolab, Pozzuoli, Italy). IRM images were acquired with a 50/50 beamsplitter (Chroma Technology, Bellows Falls, VT, USA) in the reflected light path based on a previously described method (51,63,64). Images were acquired every 1.5 min for 30 min with an Andor Clara2 CCD camera (Andor Technology, Belfast, UK). Images were background subtracted by taking a median projection of the time series, applying a Gaussian filter and subtracting this “background image” from every frame of the original time series, to enhance contrast. The 32-bit images were then converted to 8-bit for further analysis. All image processing and analysis was performed using the image processing software, FIJI ver. 2.3.0/1.53q (65).

Image quantification and statistical testing

MT lengths were estimated interactively using the line segmentation tool in FIJI by following distinct black and white lines against the gray background (Fig. S6). The overlaid images with the marked MT filaments were then quantified using the measure tool in FIJI, which was used to extract individual filament lengths and number per frame. Data used for analysis from quantitative microscopy were based on n = 3 biological replicates. For understanding the significance of difference between the mean MT lengths, we combined the mean MT lengths from each biological replicate for individual crowdant concentration. The mean MT lengths from n = 3 biological replicates was then compared between conditions of crowdants using two sampled Student's t-test using the scipy.stats.ttest_ind function in Python.

Model and simulation method

A rule-based lattice kinetic Monte Carlo model was developed to simulate linear polymer growth by the addition of monomers to the ends with diffusion of monomers and crowdants. We model four discrete entities: 1) monomers (m), 2) nucleating filaments (n), 3) polymers (p), and 4) crowdants (c). Each of these entities is modeled to have distinct microscopic events: 1) monomers undergo diffusion and association and dissociation from the polymer, 2) nucleation points are initialized to be randomly distributed throughout space, 3) polymers grow and shrink symmetrically from both ends by the addition of subunits, i.e., both ends have identical on- and off-rates, and 4) crowdants undergo diffusion. All components of the model—monomers, nucleating seeds, polymers and crowdants—follow the rule of volume exclusion, i.e., only one entity can occupy a lattice site at a given time. For simplicity of computation, monomers and crowdants are diffusive but polymers including nucleating seeds are immobile. Details of the diffusion and polymerization models are described in the following sections.

The model is simulated on a 2D square lattice in a simulation box of size $L\times L$ and spacing $\delta$ with periodic boundary conditions. Particles (monomers and crowdants can move along the basis vectors of the lattice $\left\{\left\{1,0\right\},\left\{0,1\right\}\right\}$. Monomers and crowdants are square objects with lengths $L_{monomer}$ and $L_{crowdant}$ respectively, with $L_{crowdant}$ ranging between 1/3 and 3 times that of $L_{monomer}$ (Table S1). Diffusion is modeled as an on-lattice random walk restricted to lattice sites with unit step-length $\delta$ for each molecule modeled. Diffusion of subunits and crowdants is modeled by a probabilistic, particle-specific, rate of diffusion:

$$r_{D_i}=4·D_i/{\delta }^2$$ (8)

where $D_i$ is the diffusion coefficient of the $i^{th}$ species (i = m or c), based on the Stokes-Einstein relation. On-lattice random walk is modeled as a Poisson process having exponentially distributed interarrivals (66). Random numbers are then generated that satisfy the distribution for a given diffusion rate using uniform random number by expression $E\left(r_{D_i}\right)=-\log \left(U[0,1\right))/r_{D_i}$, as described previously (67). The particle then performs random hops until the cumulative sum of interarrival times just exceeds the integration time.

Crowding is modeled by volume exclusion, i.e., no two entities in the lattice can occupy the same lattice sites simultaneously. Crowdant sizes in turn are a fold change either smaller or larger than the monomers. Their volume fraction is varied up to 0.36 based on previous reports of the percolation threshold of diffusion on a finite lattice (46).

Polymers are modeled as rigid rods that do not diffuse. Their growth is assumed to be from both ends with symmetric polymerization rates. This symmetric growth assumption evidently ignores the expected plus- and minus-end asymmetry in elongation rates of MTs. Thus, in our “toy model” the parameters are meaningful only in relation to each other and the effect of their change, not so much in absolute terms. The model also ignores the lattice structure and protofilament details, comparable with similar approaches taken in previous work (68,69,70). Elongation occurs by the probabilistic binding of monomers $\left(m\right)$ to either a preexisting nucleus $\left(n\right)$ or a polymer $\left(p\right)$ end at a rate given by $k_{pol}$ that can reversibly also dissociate at a rate $k_{depolP}$. This leads to the two mass action relations:

$$n_{\left(L_n\right)}+m\underset{k_{depolP}}{\stackrel{k_{pol}}{\rightleftharpoons }}{p}_{\left(L_n+1\right)}$$ (9)

$$p_{\left(L_p\right)}+m\underset{k_{depolP}}{\stackrel{k_{pol}}{\rightleftharpoons }}{p}_{\left(L_p+1\right)}$$ (10)

where $L_n$ is the length of the nucleating seed and $L_p$ the polymer length, both in monomer units.

Diffusing monomers are tested for interaction after every jump based on proximity to ends of nuclei or polymers. For interactions, the contact vicinity of Moore’s neighborhood of range 1 more than the monomer size is evaluated.

Model parameters were chosen to capture the relative relations between diffusion of monomers and crowdants and polymerization-depolymerization rates serving as a toy model (Table S1). Numbers of crowdant particles and their sizes were varied, resulting in changes in mobility, in order to test the effect of crowdant size and volume fraction on filament polymerization.

Simulations were performed on Intel’s Xeon Cascade Lake 2.9 GHz processors on a single processor at a time. The system RAM per node used is 192 GB using an in-house cluster https://nsmindia.in/node/157#1. Typical simulations run with 2000 monomers and $\approx$40,000 crowdants for 200 s requires 30 min. The source code (C++) is provided as supporting material.

Results

Effect of crowdant size and concentration on bulk microtubule elongation dynamics: Qualitative and quantitative effects

Single filament dynamics of MTs was observed in the presence of high viscosity solutions to result in lower MT filament elongation rates in vivo using Schizosaccharomyces cerevisiae cells and in vitro with glycerol (16). In vitro MT elongation rates were seen to slow down with increasing concentrations of small crowdants (e.g., glycerol and ethylene glycol), while large crowdants (e.g., PEG, BSA, and dextran) increased them (15). We proceeded to examine if collective polymerization, i.e., bulk turbidimetry, would also show the same crowdant size-dependent effect.

Crowdant size was approximated by the hydrodynamic radius (Table 1), which was based in turn on a range of molecular weights (Fig. 1 A). They were classified as either LMW or HMW crowdants based on a threshold MW of 1 kDa as described previously (35). LMW crowdants chosen were ethylene glycol, TMAO, glycerol, and PEG 400, some of which are considered to act as cosolvents while PEG 4000, PEG 200000, BSA, and Ficoll 400 were used as HMW crowdants, typically used in macromolecular crowding studies. The hydrodynamic radius used as measure the size of these molecules also corresponds to a threshold radius of 1 nm with LMW crowdants of MW $<1000\text{ Da}$ having a radius $r_h<1$ nm, while HMW crowdants have MW $>1000\text{Da}$ and a radius $r_h>1$ nm (Fig. 1 A, dashed lines). Some large HMW crowdants are comparable with tubulin dimers $\left(\alpha \beta \right)$ with MW $\sim$ 100,000 Da and radius $\sim$4 nm. The concentration range of each crowdant was optimized such that it resulted in comparable volume fractions (Eq. 3), based on the number density, concentration (w/v), and molecular weight, combined with the volume (Eq. 4). The volume was estimated from the hydrodynamic radius (Table 1) and a spherical approximation. Some large crowdant molecules, such as PEG 20000 with an elongated geometry, would theoretically result in unrealistic volume fractions due to an overestimation, e.g., 10% w/v of PEG 20000 would result in 100% volume fraction, assuming a hydrodynamic radius of $\sim$4 nm and spherical geometry. To avoid such artifects we use only low concentrations of PEG 20000. The spherical approximation approach is also consistent with previous work of volume fraction estimates of PEG 35 (35 kDa) and Ficoll 70 (33) as well as PEG 8000 (8 kDa) and BSA (71). Such models have previously been successfully validated by comparison to experimental data in the case of molecular mobility based on steric and nonspecific interactions (72).

Figure 1.

Effect of crowdant molecular weight and concentration on MT elongation kinetics. (A) The molecular weight of tubulin dimer (100 kDa) (green square) has been plotted against its radius (4 nm) and compared with the crowdant molecular weight and radius. The crowdants have been classified using a threshold of 1000 Da into LMW crowdants (open symbols)—ethylene glycol, TMAO, glycerol, PEG 400, and HMW (solid symbols) crowdants—PEG 4000, PEG 20000, BSA, and Ficoll 400, as used in (35). (B) The schematic depicts the steps involved in MT seed nucleated polymerization kinetics with sheared, GMPCPP-stabilized MTs serving as seeds for GTP-dependent tubulin polymerization (elongation) in the presence of crowdants measured as absorbance kinetics (details in the material and methods). (C–J) The polymerization kinetics of 8 $\mu$M goat brain tubulin with 1 mM GTP in the presence of GMPCPP-stabilized MT seeds and crowdants are measured in terms of the absorbance at 340 nm over time (min) after blank subtraction (Fig. S1, as described in the material and methods). Crowdants used are: LMW crowdants (C) ethylene glycol, (D) TMAO (75 Da), (E) glycerol (92 Da), and (F) PEG 400 (400 Da); and HMW crowdants: (G) PEG 4000 (4 kDa), (H) PEG 20000 (20 kDa) (I) BSA (66 kDa), and (J) Ficoll 400 (400 kDa). Crowdant concentration: % w/v (colors); circles: mean absorbance after blank subtraction; blank: tubulin in the presence of crowdants without seeds (Fig. S1); shaded region: standard deviation (n = 6), solid lines: fit to the NDP model (Eq. 1).

To measure MT polymerization through elongation, we assembled GMPCPP-stabilized MT seeds from 45 $\mu$M goat brain tubulin in the presence of 0.8 mM GTP, to serve as nucleators. These were then combined with 8 $\mu$M goat brain tubulin, GTP, ions, and crowdants and the polymerization kinetics measured by absorbance at 340 nm over time (Fig. 1 B). The polymerization in the absence of nucleators, (−)seeds, was also measured as a reference (Fig. S1). The concentration of subunits used, 8 $\mu$M tubulin, is below the critical concentration as demonstrated by the lack of polymerization even in the presence of nucleators, (+)seeds, unless a crowdant was added, such as 8.3% ethylene glycol (Fig. S2). We estimated the crowdant-free critical concentration from absorbance kinetics of increasing concentrations of tubulin (Fig. S3 A) fit to the NDP model (Eq. 1) to estimate the polymerization rate, $r$. The data of increasing $r$ with concentration fit to a linear function provides a measure of the crowdant-free critical concentration, c^∗ $\approx$ 12.1 $\mu$M (Fig. S3 B).

The elongation kinetics were then measured as the difference between absorbance with nucleators and without (Fig. S1), for multiple crowdants ranging in concentration between 0.5 and 31.5% (% w/v): 1) LMW crowdants—ethylene glycol (MW = 62.07 Da, Fig. 1 C), TMAO (MW = 75 Da, Fig. 1 D), glycerol (MW = 92 Da, Fig. 1 E), and PEG 400 (MW = 400 Da, Fig. 1 F)—and 2) HMW crowdants—PEG 4000 (MW = 3.5–4.5 kDa, Fig. 1 G), PEG 20000 (MW = 3.5–4.5 kDa, Fig. 1 H), BSA (MW = 66.43 kDa, Fig. 1 I), and Ficoll 400 (MW = 400 Da, Fig. 1 J).

Increasing concentrations of crowdant affect polymerization both in terms of steady-state absorbance at saturation, reflecting the polymer mass fraction and rate of increase in absorbance, a measure of polymerization rate. Comparing the qualitative differences between the large and small crowdants, we find that small crowdants demonstrate a concentration-dependent decrease in both polymerization rate and steady-state polymer mass as seen in the case of ethylene glycol, TMAO, glycerol, and PEG 400 (Fig. 1, C–F). Large crowdants on the other hand increase the rates, as seen in the case of PEG 4000, PEG 20000, BSA, and Ficoll 400 (Fig. 1, G–J). A quantitative perspective on these effects is obtained by fitting a NDP kinetic model (Eq. 1) to the absorbance kinetic data, as described previously (51,73). The model is used to estimate three parameters: the normalized steady-state absorbance $[A_{\max }$ − $A_0$] (where $A_{\max }$= steady-state absorbance; $A_0$ = initial value), the polymerization rate $r$, and half-maximal time of polymerization $t_{1/2}$. Consistent with the qualitative observations, and the difference in the size dependence of crowdant concentration, we find LMW crowdants result in decreasing steady-state absorbance and growth rates while increasing the $t_{1/2}$ (Fig. S7, A–D). These trends quantitatively suggest an inverse relation of polymerization kinetics with LMW crowdant concentration—with $t_{1/2}$ increase capturing the delay time increase, the rate the speed of assembly, and maximal absorbance captures polymer mass fraction. In contrast, HMW crowdants result in a concentration-dependent increase in the relative steady-state absorbances and polymerization rates and a corresponding decrease in the delay to reach maximum, i.e., half-maximal time (Fig. S7, E–H).

Thus, we find that “seed-based” polymerization monitored using optical scattering shows that the rate and steady-state polymer mass decrease with the concentration of LMW crowdants, but increase with HMW crowdants. Our observations using collective polymerization are consistent with previous observations at the level of single-filament dynamics (15). The reduced elongation rate effect of small crowdants had been attributed to a concentration-dependent increase in viscosity (16). To examine how LMW and HMW crowdants affect viscosity we proceed to measure this.

Crowdant size and concentration dependence of viscosity measured by micron-scale bead diffusion

To address the question how viscosity varies with crowdant size and concentration, we compared the effect of increasing crowdant concentration on the mobility of tracer particles of 1 $\mu$m diameter. The microscopic mobility of these beads was recorded in time-lapse microscopy in a $\sim$2 $\mu$L volume and quantified by single-particle tracking using a previously developed image analysis tool (62). The XY-trajectories over time were used to calculate MSD ($⟨r^2⟩$) plots over increasing time intervals and the linear region of the graph fit to the expression: $⟨r^2⟩=4·D_{eff}·t$, corresponding to 2D diffusion. The free parameter was the effective diffusion coefficient ($D_{eff}$), which was thus estimated for increasing crowdant concentration and showed an expected decrease (Fig. S8, A and B, top). The viscosity $\eta$ of the solutions was estimated from Stoke-Einstein’s relation and showed a concomitant increase with concentration (Fig. S8, A and B, bottom). However, the decrease in diffusion coefficient and increase in viscosity is similar in terms of concentration dependence, independent of the size of the crowdant. This suggests that the size-dependent difference in the effect of crowdant concentration on MT elongation rate does not arise from the bulk viscosity, but a scale-dependent effect. Such scale-dependent diffusive mobility has been reported earlier too, where the probe size determines diffusive mobility (27). Smaller crowdants, of the scale of $⟨1$ nm in diameter, are expected to act as viscogens, increasing viscosity and decreasing diffusivity, but the large crowdants ($r_h>1$ nm) do not affect local diffusivity over short ranges, but instead create microcompartments, increase effective concentration, and enhance rates—the excluded volume effects (EVEs), as described in the case of actin polymerization (14). We proceeded to test this using a lattice Monte Carlo simulation.

Kinetic Monte Carlo model predicts role of microviscosity in reduced elongation in the presence of small crowdants

A stochastic, discrete, particle, lattice-based 2D computational toy model of monomer and crowdant diffusion combined with polymerization was developed. The approach adopted is comparable with previously developed lattice rule-based models of diffusion and aggregation (50,74,75). In addition, the model is dimensionless in space, and abstract at the level of polymer description, ignoring detailed protofilament structure. For the model the spatial scale is set by the lattice spacing, of unit size. Previously described simulations on the regulation of dynamic instability (68), the role of random GTP hydrolysis in cap stability of growing phase MTs (69), and kinetic models of dynamic switching transitions (70) have used similar abstract descriptions of MTs to model polymerization dynamics. In the model, mobility is simulated based on size of the particle and volume exclusion, i.e., no two objects can occupy the same position in the lattice. For simplicity, filaments are assumed to be immobile, linear, rigid rods that grow only from their ends. Furthermore, for simplicity the elongation of nucleating seeds is assumed to be symmetric from both ends, ignoring the plus- and minus-end dynamics of MTs. In the toy model the size of the crowdants was mapped to relative lattice length scale with $L_{monomer}$ = 3, while crowdants occupied squares with side lengths $L_{crowdant}$ of 1, 3, 5, 7, or 9 (Table S1). We examined the effect of increasing crowdant size and volume fraction, i.e., relative area occupied by the crowdant and found that diffusive trajectories are qualitatively more confined in the presence of small crowdants compared with longer traces in the presence of large crowdants (Fig. S9). We quantified this by calculating the time-averaged TAMSD and fitting it to the anomalous diffusion model (Fig. 2 A) to arrive at an estimate of the diffusion coefficient $\left(D\right)$. The resulting estimates of the relative diffusion coefficient $\left(D/D_0\right)$ show a pronounced effect in the presence of small crowdants compared with large crowdants (Fig. 2 B). $D_0$ is the input diffusion coefficient. The exponent of anomalous diffusion, $\alpha$, is seen to decrease from 0.98 to 0.9 with increasing volume fraction of all crowdants (Fig. 2 C), with crowdant size increase having a weaker effect on $\alpha$. This appears to suggest that the effect of anomalous diffusion is small, with the predominant effect in terms of ${D/D}_0$, relative diffusion coefficient, and in turn due to microviscosity.

Figure 2.

Simulating the effect of crowdant size and density on monomer diffusion and filament polymerization. (A) Simulated XY-trajectories of monomers (size 3 × 3) were used to calculate the time-averaged mean-square displacement (TAMSD) $\left(⟨r^2⟩\right)$ in lattice ${\text{unit}}^2$/s and plotted on a log-log plot as a function of time interval (x axis). Crowdant sizes, $L_{crowdant}$ (left to right): 1, 3, 5, 7, and 9 lattice units, while volume fractions ranged from 0 to 0.36 (color bar). The solid short lines denote regions used for fitting. (B and C) The effect of increasing crowdant volume fraction ${\varphi }_c$ (x axis) was examined by fitting the anomalous diffusion model to the TAMSD by the function $4D{t}^{\alpha }$ and quantifying (B) the ratio $D/D_0$ (y axis), where D is the diffusion coefficient fit with $\alpha$ = 1 and $D_0$ is the input diffusion coefficient and (C) $\alpha$ (y axis) the exponent of anomalous diffusion, obtained from fitting the TAMSD profile with both D and $\alpha$ as fit parameters. Here, $D_0$ = 18.125 $l{u}^2$/s. (D) Kinetics of polymer fraction (y axis) increase as a function of time (x axis) in the presence of nucleating “seeds.” Left to right: $L_{crowdant}$: 1, 3, 5, 7, and 9 lattice units. Color bar: area fraction of the crowdant. (E and F) The last 5 s of polymerization kinetics simulations were averaged to obtain the (E) fractional polymer mass and (F) the filament lengths $\left(⟨L_{MT}⟩\right)$. (G) The schematic represents the differences in monomer mobility (blue box) due to different sizes of crowdants (magenta boxes), for the same ${\varphi }_c$. All statistics are averaged over n = 5 runs. The shaded areas represent $\pm$ SD. Total simulation time for diffusion = 20 s and polymerization = 200 s.

Polymerization kinetics simulated in the presence of preexisting nucleators or seeds $\left(N_s\right)$ and freely diffusible subunits show a prominent decrease in total polymer fraction with increasing crowdant volume fraction $\left({\varphi }_C\right)$ and small crowdants ($L_{crowdant}$ = 1 and 3) (Fig. 2 D, left), while large crowdants ($L_{crowdant}$ = 5, 7, 9) showed very small changes, in polymeric fraction at saturation (Fig. 2 D, right). This is also seen in representative visualization of the simulation outputs for the smallest crowdant ($L_{crowdant}$ = 1) and largest crowdant ($L_{crowdant}$ = 9), each at volume fractions $\left({\varphi }_C\approx 0.23\right)$ where the initial length of nucleating seeds, their growth into filaments, and the diffusion of monomers and crowdants is observed (Fig. S10; Videos S1A and S1B). The steady-state polymer fraction shows a prominent decrease with crowdant volume fractions for small ($L_{crowdant}$ = 1), but little or no change for all other crowdants from $L_{crowdant}$ = 3 to 9 (Fig. 2 E), which is due to reduced elongation as seen in the mean MT lengths (Fig. 2 F) and the frequency histogram of crowdant size and density dependence (Fig. S11). These results are consistent both with our experimental kinetics of MT elongation described in the previous section in terms of the effect of small crowdants, However, possibly due to the weaker effect of the EVE we do not observe the enhanced polymerization observed experimentally.

In summary, our simulations suggest a decline in relative diffusion coefficients (${\mathit{D}\mathit{/}\mathit{D}}_0$) of monomers with increasing concentration of small crowdants, while crowdant size increases appears to reduce this effect. Thus microviscosity appears to be modified by the same volume fraction of small crowdants, potentially since they hinder mobility of subunits uniformly, while large-sized crowdants for the same volume fraction result in inhomogeneity in space with areas devoid of crowdants resulting in comparatively free mobility of monomers (Fig. 2 G).

These simulations demonstrate that the diffusion-limited nature of elongation could explain the crowdant size dependence of polymerization from seeds. MTs in the cellular context are likely to also spontaneously nucleate based on the reported in vivo tubulin cellular concentration of $\sim$2 mg/mL (76,77), since it exceeds the critical concentration. We thus proceeded to examine how the combined process of nucleation and elongation are affected by crowdant size and concentration.

Crowdant concentration increases de novo polymerization independent of crowdant size

Microtubule polymerization is nucleation dependent and involves two steps, a rate-limited MT nucleation (78) and elongation, together referred to as nucleation-dependent polymerization (NDP) (51,59,73,79). We proceeded to test the effect of crowdants on both nucleation and elongation by polymerizing 30 $\mu$M tubulin in the presence of 1 mM GTP and increasing concentrations of crowdants without the addition of seeds, as schematically depicted (Fig. 3 A), and referred to in previous work as de novo polymerization (80). We observe a common effect of decreased lag periods and increased steady-state absorbance with increasing crowdant concentration in LMW crowdants: ethylene glycol (Fig. 3 B), TMAO (Fig. 3 C), glycerol (Fig. 3 D), and PEG 400 (Fig. 3 E), as well as HMW crowdants PEG 4000 (Fig. 3 F), PEG 20000 (Fig. 3 G), BSA (Fig. 3 H), and Ficoll 400 (Fig. 3 I).

Figure 3.

Effect of crowdant molecular weight and concentration on the kinetics of de novo MT polymerization. (A) The schematic represents how polymerization of $\alpha$-,$\beta$-tubulin dimers (green and purple circles) was followed in the presence of crowdants (red circles) by absorbance kinetics. The experiments were conducted without the addition of nucleating seeds. (B–I) The absorbance at 340 nm ($A_{340nm}$) is plotted as a function of time in minutes, as a measure of the polymerization of 30 $\mu$M goat brain tubulin with 1 mM GTP, with increasing concentrations (% w/v) of LMW crowdants: (B) ethylene glycol, (C) TMAO (75 Da), (D) glycerol (92 Da), and (E) PEG 400 (400 Da); and HMW crowdants: (F) PEG 4000 (4 kDa), (G) PEG 20000 (20 kDa) (H) BSA (66 kDa), and (I) Ficoll 400 (400 kDa). Polymerization in the absence of crowdant (0%) (black) with 30 $\mu$M tubulin is repeated in (Fig. S3A) for comparison. Polymerization for a range of tubulin concentrations without crowdants are described in Fig. S3A. The plot depicts mean absorbance (colored circles) averaged for three replicates and combined with the standard deviation of each biological replicate (shaded region). Solid lines: fit line based on the NDP model (Eq. 1). Colors: crowdant concentrations.

Similar to the elongation rate estimation, we quantified the dynamics of de novo polymerization by fitting the absorbance kinetics to the NDP model (Eq. 1) to obtain the normalized steady-state absorbance $[A_{\max }$ − $A_0$], polymerization rate $\left(r\right)$, and time to reach half-maximal absorbance ($t_{1/2}$). All crowdants are observed to increase steady-state absorbance ($[A_{\max }$ − $A_0$]) and polymerization rates, $r$, in a concentration-dependent manner—LMW: ethylene glycol (Fig. S12 A), TMAO (Fig. S12 B), glycerol (Fig. S12 C), and PEG 400 (Fig. S12 D); and HMW: PEG 4000 (Fig. S12 E), PEG 20000 (Fig. S12 F), BSA (Fig. S12 G), and Ficoll 400 (Fig. S12 H). Correspondingly the value of ${\mathrm{t}}_{1/2}$ shows a decreasing trend with concentration (Fig. S12, A–H, bottom). This demonstrates that, irrespective of crowdants size, whether LMW or HMW, the polymerization kinetics increase in a concentration-dependent manner during de novo polymerization in bulk. The effects and fold differences are observed over different concentration ranges for the small and large crowdants, and are related to size and the volume fraction, which we further investigate.

We hypothesize that all crowdants increase the nucleation rate, which could explain the difference between the size-dependent effect of crowdants on elongation but not on de novo polymerization.

Size-independent effect of crowdant concentration on nucleation

To test the hypothesis that the differences in crowdant size effects on elongation and de novo polymerization can be explained by the effect on nucleation, we measure the critical concentration (c^∗) of tubulin as a readout of MT nucleation (1,51,81). We measured the kinetics of de novo polymerization over a range of tubulin concentrations between 10 and 30 $\mu$M, combined with multiple concentrations of two representative small and large crowdants—LMW: ethylene glycol and glycerol; and HMW: PEG 4000 and Ficoll 400. The polymerization rate plotted as a function of tubulin concentration for each concentration of crowdant was fit to straight line functions to obtain the critical concentrations, as described in the material and methods (Fig. 4, A–D). Increasing the crowdant concentration results in decreases in critical concentration in the case of ethylene glycol (Fig. 4 E), glycerol (Fig. 4 F), PEG 4000 (Fig. 4 G), and Ficoll 400 (Fig. 4 H), i.e., for both small (LMW) and large (HMW) crowdants. The critical concentration of brain tubulin reduced by a factor of 2 over the range of concentrations tested of both LMW and HMW (Table S2) compared with c^∗ of 12.1 $\mu$M without the addition of crowdants (Fig. S3 B). This demonstrates that MT nucleation is enhanced by increasing crowdant concentration, independent of the crowdant size (Fig. 1 A). The normalized steady-state absorbance ($[A_{\max }$ − $A_0$]), a measure of relative polymer mass at steady state, was also used to measure critical concentration, using a similar fitting-based approach (Fig. S13, A–D) and resulted in similar values and trends in critical concentration with increasing crowdant concentration (Fig. S13, E–H; Table S2). Critical concentration values less than zero are ignored.

Figure 4.

Effect of crowdants on critical concentration of tubulin polymerization based on the polymerization rate $r$. (A–D) The polymerization rate, r (y axis) for increasing tubulin concentrations (x axis) is plotted with increasing (A) ethylene glycol, (B) glycerol, (C) PEG 4000, and (D) Ficoll 400. The values were then fit by a straight line model (Eq. 2). The x-intercept of the line is the critical concentration c^∗ (vertical dashed lines). Circles with error bars: mean $\pm$ SD (n = 6); colors: crowdant concentration. (E–H) The resulting mean critical concentration values (y axis) from n = 6, are plotted against different concentrations (% w/v) (x axis) of the crowdants (E) ethylene glycol, (F) glycerol, (G) PEG 4000, and (H) Ficoll 400. The critical concentration for no crowdant conditions (0%, black) were estimated from de novo polymerization of 15–30 $\mu$M tubulin in the absence of crowdants (Fig. S3A), as shown in Fig. S3B, and compared for each crowdant.

However, these findings of nucleation and elongation dependence on crowdant size are based on indirect, turbidimetric assays, i.e., they measure total polymer mass. To reconcile our findings about elongation being differentially affected by crowdants based on their size, but size-independent nucleation, we proceeded to examine de novo MT polymerization at a single filament level.

LMW crowdant concentration increases MT filament density but decreases MT lengths

Single-filament dynamics are typically measured one filament at a time and have provided deep insights into regulation of MT polymerization. Here, to better understand our bulk polymerization kinetics measurements, we aimed to measure collective dynamics of simultaneous nucleation and polymerization. Tubulin samples with GTP and crowdants were added to double-backed tape chambers and sealed between a slide and coverslip (Fig. S5 A). We observed the emergence of MT filaments from free subunits (nucleation) and the growth of those filaments (elongation) with time in the presence of increasing concentrations of LMW crowdants—ethylene glycol and glycerol or HMW crowdant—PEG 4000 in time lapse label-free IRM (Fig. S5 B) —as described in detail in the material and methods. For all three crowdants, concentration increase resulted in increased nucleation, as seen in the increased filament density (Fig. 5, A–D, and G). While representative images show snapshots after 12 min of incubation, due to the observed time for steady state in bulk kinetics, time lapse microscopy of the samples reveals that ethylene glycol in low concentrations (8.3%) leads to the formation of distinct and long filaments with filament numbers increasing with time, while a higher concentration (22.22%) results in more filaments being formed but of shorter lengths (Fig. S14 A). A similar trend of shorter but more numbers of filaments is observed in time series of tubulin with 6.3–31.5% glycerol (Fig. S14 B). PEG 4000 (2–4%), while showing increase in filament numbers, has little visible difference in lengths (Fig. S14 C). Quantification of MT lengths from IRM time series reveals the decrease of MT lengths with increasing concentration of both ethylene glycol (Fig. 5 B) and glycerol (Fig. 5 E), but not significantly for PEG 4000 (Fig. 5 H). This is consistent with our examination of crowdant size dependence on elongation. While the absolute concentration ranges of the crowdants differ, the volume fractions calculated from their hydrodynamic radii (Table 1) are of similar ranges (Fig. S15), suggesting that the volume fraction is driving the observed effects. The quantification of IRM was performed by overlaying outlines of bona fide filaments and measuring their lengths converted from pixels to micrometers (Fig. S6 and material and methods). In the absence of crowdants, we find that microscopy does not reveal any observable nucleation or elongation of filaments even after 12 min.

Figure 5.

Effect of crowdant concentration and size on MT length distribution. (A, D, and G) Representative images from an IRM (label-free microscopy) time series of 20 $\mu$M tubulin with 1 mM GTP polymerization in a coverslip-chamber in the presence of (A) 8.3, 11.11, and 22.22% (w/v) of ethylene glycol, (D) 6.3, 18.9, and 31.5% (w/v) of glycerol, or (G) 2, 4, and 6% (w/v) of PEG 4000 after 12 min. Scale bar, 2 $\mu$m. (B, E, and H) The MT length distribution in the presence of increasing concentrations of (B) ethylene glycol, (E) glycerol, and (H) PEG 4000. Horizontal lines denote means for individual biological replicates. The differences in mean MT lengths from all the biological replicates across concentrations were tested for significance using Student's t-test with ^∗∗∗∗p$⟨0.0001$, ^∗∗∗p$⟨0.001$, ^∗∗p$⟨0.01$, ^∗p$⟨0.05$; ns, not significant. ${\varphi }_c$: volume fraction for individual crowdant concentrations. (C, F, and I) Effect of concentrations of (C) ethylene glycol, (F) glycerol, and (I) PEG 4000 on (i) MT length and (ii) MT number/100 $\mu {\mathrm{m}}^2$ respectively. Circles: mean $\pm$ SD for three individual biological replicates.

The quantification of mean MT lengths ($L_{MT}$) and numbers ($N_{MT}$) show that small-sized crowdants result in significantly shorter but higher filament density MTs for increasing concentrations of ethylene glycol (Fig. 5, B and C) and glycerol (Fig. 5, E and F)—supporting the reports that small molecules act to reduce elongation by diffusion limitation. A large crowdant such as PEG 4000 does not alter MT lengths significantly (Fig. 5, H and I(i)) but does increase the density (Fig. 5 I(ii)).

Filament density of de novo polymerized MTs quantified from microscopy appears to increase uniformly with volume fraction independent of crowdant size (Fig. 6 A), while MT lengths appear to decrease with increasing volume fraction for ethylene glycol or glycerol (small), but remain unchanged with PEG 4000 (Fig. 6 B). In simulations two representative packing fractions of ${\varphi }_C\approx 0.23$ and 0.36 result in qualitatively longer filaments in the presence of large ($L_{crowdant}$ = 9) as compared with small ($L_{crowdant}$ = 1) crowdants, for a constant number of nucleators (Fig. 6 C). Simulations of a range of crowdants predict that the normalized mean lengths, i.e., the ratio of the mean MT lengths $\left(L_{MT}\right)$ to the maximal length observed $\left(L_{\max }\right)$, decreases with increasing packing fraction only for small crowdants $\left(L_{crowdant}\right)$ but remains largely unaffected for all other crowdant sizes simulated in our toy model (Fig. 6 D, connected points). To our surprise, the experimentally measured relative mean lengths from de novo polymerization in the presence of increasing packing fraction of ethylene glycol, glycerol, and PEG 4000, are qualitatively comparable with simulations (Fig. 6 D, symbols). In simulations, MTs grow only by seed-based elongation, while experiments involve both nucleation and elongation. The match between experiments and simulations suggests nucleation, which is rate limited, and elongation, which is diffusion limited, respectively, govern MT polymerization in a manner independent of one another.

Figure 6.

MT lengths and number density dependence on volume fraction from experiments and simulations. (A–C) Increasing crowdant volume fraction (${\varphi }_c$, x axis) dependence of (A) MT lengths and (B) filament density per 100 $\mu {\mathrm{m}}^2$ (${\rho }_{MT}$) are plotted. Solid symbols: small (LMW); empty symbols: large (HMW), values are mean $\pm$ SD (n = 6). The MT density quantified from 1) ethylene glycol: n = 308 (8.3%), n = 715 (11.11%), and n = 1424 (22.22%), 2) $\blacksquare$ glycerol: n = 67 (6.3%), n = 590 (18.9%), and n = 1326 (31.5%), and 3) PEG 4000: n = 578 (2%), n = 989 (4%), and n = 1653 (6%). (C) Representative outputs of simulations of MT lengths for two volume fractions—${\varphi }_C\approx 0.23$ (top) and ${\varphi }_C\approx 0.36$ (bottom) with either small ($L_{crowdant}=1$) (left) or large crowdants ($L_{crowdant}=9$) (right) as observed at t = 20 s. Red squares, crowdants; green squares, free subunits; blue lines, polymers. (D) The normalized mean MT lengths for each crowdant (${\mathrm{L}}_{MT}$/ ${\mathrm{L}}_{\max }$) from microscopy (symbols, left y axis) are compared with simulations (dashed lines, right y axis) as a function of ${\varphi }_C$. Colors: $L_{crowdant}$ = 1 to 9 lattice units. Arrows: ${\varphi }_C\approx$ 0.23 and 0.36, corresponding to simulation snapshots in (C).

Interestingly, the elongation rate dependence on viscosity varies with crowdant size: for large crowdants it increases and can be fit by a saturation function (Eq. 6), while for small crowdants it decreases and can be fit by an inverse viscosity dependence (Eq. 7) (Fig. 7 A). De novo polymerization rates from turbidimetry (bulk), however, all increase linearly, with some crowdants showing very little change (Fig. 7 B). This distinction between the effects on change in elongation and de novo polymerization rates is also evaluated in terms size of the crowdants. The change in rates of polymerization between highest and lowest crowdant concentrations, $\Delta r_{elong}$, is negative for LMW and positive for HMW crowdants based on the 1 kDa threshold (Fig. 7 C). A similar measure for de novo polymerization rates, $\Delta r_{denovo}$, is positive, irrespective of crowdant size or threshold determining whether a crowdant is LMW or HMW (Fig. 7 D). The polymerization rate change with concentration of crowdants also depends on the hydrodynamic radius, with $\Delta r_{elong}$ negative when the hydrodynamic radius $r_h<1$ nm, i.e., small crowdants, and positive when the radius $r_h>1$ nm, i.e. large crowdants (Fig. 7 E). Consistent with the effect of the cutoff MW, de novo polymerization rate changes $\Delta r_{denovo}$ are positive irrespective of hydrodynamic radius of the crowdants (Fig. 7 F).

Figure 7.

Dependence on crowdant viscosity and size on bulk MT elongation rate and de novo polymerization rates. (A and B) Viscosity, measured from microrheology (x axis) using either small (LMW, solid symbols) or large (HMW, empty symbols) crowdants, is correlated to (A) MT elongation rates, $r_{elong}$ and (B) de novo polymerization rates, $r_{denovo}$. The large (HMW) crowdant data was fit to a saturation function (Eq. 6) and the small (LMW) to $r=c_1/\eta$ (Eq. 7). The de novo polymerization rates were fit, for each crowdant, to a straight line: $r_{denovo}=c_2·\eta +c_3$. $c_2$ and $c_3$ are fit parameters. ${\mathrm{R}}^2\ge$ 0.92. Symbols: mean $\pm$ SD (n = 6). (C–F) $\Delta r=r\left(\left[C_{\max }\right]\right)-r\left(\left[C_{\min }\right]\right)$ quantifies the difference between polymerization rates from the maximal (r$\left(\left[C_{\max }\right]\right)$ and minimal concentrations (r$\left(\left[C_{\min }\right]\right)$ of the respective crowdants for (C and E) bulk elongation, $r_{elong}$ and (D and F) de novo polymerization rates, $r_{denovo}$, as a function of (C and D) crowdant molecular weight (Da) (x axis) and (E and F) hydrodynamic radius (nm) (x axis) for both LMW (solid symbols) and HMW (open symbols) crowdants.

Thus, we can show that de novo polymerization in the presence of crowdants that range in size appears to be the combined result of the size-independent effect of crowdants on nucleation, and seed based (elongation) and that the two processes appear to coexist.

Discussion

Here, we have used a combination of experiment and simulation to test the effect of crowdant size and concentration on MT polymerization. We find that elongation rates in bulk kinetics are reduced by LMW (small) crowdants while HMW (large) crowdants increase them in a concentration-dependent manner. Experimental viscosity measurements of micron-sized probes show a uniform increase in viscosity with crowdant concentration, irrespective of size. However, kinetic Monte Carlo simulations of the effect of crowdant density show that 1) small crowdants decrease the diffusion of subunits more dramatically than large crowdants, with increasing concentration, 2) the small crowdants increase microviscosity, 3) mean MT lengths of elongating filaments decrease with increasing small crowdant concentration but remain unchanged for large crowdants, and 4) the volume fraction of crowdants ${\varphi }_C$, fraction of space occupied is key to understanding the effect on diffusion and polymerization. In experiments in the absence of seeds, i.e., de novo polymerization, we find crowdant concentration decreases the tubulin critical concentration, c^∗, i.e., enhances nucleation, and the total polymer fraction increases, independent of crowdant size. Microscopy time series analysis of de novo polymerization demonstrates crowdant size-independent increase in polymer fraction with crowdant concentration can be explained by the combined effect of crowdants size-independent increase in nucleation resulting in the formation of more MT ends, and size-dependent effect on elongation, which reduces MT lengths when crowdants are smaller than the subunits and do not affect length when they are larger. The volume fraction of crowdants in de novo polymerization of large and small crowdants allows us to contrast the effect of crowdant size on MT length (Fig. 6 A) and number (Fig. 6 B). We show that viscosity dependence of collective elongation rates $r_{elong}$ (${\min }^{-1}$) are crowdant size dependent (Fig. 7, C–E), but de novo polymerization rates are not (Fig. 7, D–F).

Microtubule nucleation and elongation are highly regulated in cells by a range of microtubule-associated proteins that continue to be discovered (reviewed by Bodakuntla et al. (8)) as well as the “tubulin code” of post translational modifications (reviewed by Janke and Magiera (82) and Roll-Mecak (83)). At the same time, the physical properties such as crowding also affect microtubule polymerization, and are important since the cell is a crowded environment. Previous studies had established that the critical concentration of brain tubulin polymerization could be reduced from the $\sim$10 $\mu$M range under standard conditions (59) to 2.5 $\mu$M on addition of crowdants such as PEG 6000 and dextran-T10 (84). Glycerol was also shown to increase the rate of initial MT assembly and slowing depolymerization (85). The effect of small crowdants to decrease MT elongation rates of microtubules while large crowdants showing increased elongation has been reported before (15) and discussed in terms of in vivo viscosity increases (16). However, as we can show, the viscosity measured using micron-sized spheres increases irrespective of crowdant size (Fig. S8). Simulations of relative scale-dependent viscosity, show that small crowdants of a size smaller than the polymerizing subunits have a larger effect with increasing concentration on diffusivity as compared with crowdants of larger size (Fig. 2 B). This is referred to as microviscosity based on a change of the diffusion coefficient itself without subdiffusive motion (14,86,87). A divergence of polymerization rates due to increasing bulk viscosity on polymerization, depending on crowdant size, had been previously been reported for actin elongation rates in the presence of LMW while it leads to an increase in elongation rates if the crowdant is HMW (13). This can be resolved by ignoring bulk viscosity and considering the spatial structure that crowdants generate on the scale of the subunits, with small crowdants resulting in smaller pore sizes compared with large crowdants (88). In the case of actin this has been demonstrated to increase microviscosity in the case of small crowdants thus reducing polymerization rates while large crowdants act through volume exclusion to enhance actin polymerization (14). In our study we demonstrate using lattice-based diffusion simulations that indeed mobility is reduced by LMW but not HMW crowdants, and this is consistent with the diffusion-limited nature of elongation.

TMAO, which is a natural osmolyte has been shown in a number of studies to promote protein stability (89,90,91). We find that TMAO decreases MT elongation rates, i.e., polymerization in the presence of MT seeds, in a concentration-dependent manner (Figs. 1 D and S7 B). This appears to contradict previous work that demonstrated that TMAO serves to promote MT assembly and counteracts the inhibition of polymerization by urea (58,92). However, de novo polymerization in the presence of increasing TMAO concentrations does still show a similar effect of polymerization enhancement measured by turbidimetry (Figs. 3 C and S12 B). This further supports our conclusion, that TMAO acts in a manner similar to small crowdants or cosolutes by enhancing MT nucleation, independent of the effect on diffusion-limited elongation. This effect is similar to that of glycerol, where increasing concentrations result in a higher viscosity (Fig. S8), thus reducing the diffusion coefficient and reduced rates of seed-based MT elongation. Our results highlight the generality of the results and could help to reconcile potentially contradictory findings.

In a study of the effect of osmolytes and crowding on MT polymerization, a range of 10–20% w/v of sucrose (MW = 342.30 Da) was observed to have very little effect on the de novo polymerization kinetics of tubulin (58). Since the radius of gyration of sucrose is reported to be 0.4 nm (93), based on our model we expect it to act as a viscogen. In such a case, increasing concentration should lead to increase in polymerization kinetics rate. We believe the reason for the lack of any effect is the range of concentrations used that correspond to volume fractions ${\varphi }_C$ ranging between 0.05 and 0.09. These are likely to be too small to have a significant effect on polymerization rates based on our observations and predictions. Thus, we can identify a range of volume fractions that are predicted to measurably affect polymerization of tubulin. This could help better understand both in vitro and in vivo effects observed.

The simulations described here predict a decrease in elongation of MTs with increasing LMW crowdants, consistent with experiments. However, in simulations we do not observe the increase in elongation as a result of increasing HMW crowdant concentration. While bulk experiments (collective polymerization) suggest that large crowdants result in an increased elongation rate with concentration (Fig. S7), microscopy data for one large crowdant suggests that filament lengths themselves do not change with large crowdants (Figs. 5, G–I and 6 A). In the past, explanations for increased rates with crowdants have been attributed to the excluded volume effect, EVE (42,94), which serve to increase local concentrations, for example, during actin polymerization (14). A possible explanation for the lack of EVE-induced rate increases in simulations, which we do observe in experiment, could be due to structure formation of crowdants; this is not explicitly modeled. In the future, we will seek to further explore models of corrals to represent microdomains (50). At the same time, our calculations have allowed for the interpretation of polymerization and, with the addition of multistep nucleation kinetics (73), could be used to examine more complex in vivo scenarios.

Our experiments confirm that MT nucleation appears to be rate limited, while elongation is diffusion limited. This is similar to reports from actin polymerization (13). Thus, MT nucleation showing lowering of critical concentrations in the presence of small and large crowdants can be understood to arise from EVEs, while elongation is reduced when viscosity increases due to small crowdants, similar to the observations with actin (13,14) and amyloid fibrils (42). At the same time we demonstrate that bulk viscosity increases as a function of crowdants irrespective of size for a comparable range of volume fractions. It is the role of LMW crowdants in increasing the microviscosity that leads to decrease in elongation rates by affecting monomer diffusion. This appears to make the interpretation of the in vivo data all the more puzzling, since Molines et al. (16) have reported a decrease in MT elongation rates in Schizosaccharomyces pombe cells due to osmotic increase in intracellular crowding, suggesting that the crowding is primarily influencing microviscosity. It would be useful to test whether mixed crowdants in vitro would show a dominance of the small over large crowdants to possibly resolve this question.

Conclusion

We conclude that crowdant size and concentration influence both MT nucleation and elongation and that our model demonstrates the effect of crowding on diffusion-limited MT elongation, while the effect on de novo polymerization can be understood as a collective outcome of the increase in nucleation independent of crowdant size, with a size-dependent change in elongation rates. This would could help better understand the role of such collective physical effects on the growth and regulation of cytoskeletal polymers in cell physiology.

Acknowledgments

We are grateful to Shivani Yadav for advice on tubulin purification and polymerization experiments and Thorsten Wohland and Fred Chang for discussions. J.B. is supported by a PhD studentship from IISER Pune. A.S. is supported by a fellowship from the Department of Biotechnology, Govt. of India (DBT/2021-22/IISER-P/1851). The project was supported by IISER Pune core funds to C.A.A. The support and the resources provided by PARAM Brahma Facility under the National Supercomputing Mission, Government of India at the Indian Institute of Science Education and Research Pune (IISER Pune) are gratefully acknowledged.

Author contributions

J.B. carried out all experiments, data analysis, made the figures, and wrote the manuscript. A.S. carried out all simulations and made the figures. C.A.A. designed and supervised the research, obtained funding, and wrote the article.

Declaration of interests

The authors declare they have no conflict of interest.

Editor: William Hancock.

Supporting material

Data S1. Supporting source code

The source code used to produce the simulations described in the manuscript.