The formation of biomolecular condensates driven by LLPS in living systems often require adequate interactions between biomolecules, subcellularly concentrated biomolecules, and multivalent biomolecular interactions to maintain their dynamic properties. In this work, we developed and simulated a stochastic model in silico using the LAMMPS44,45 to understand the role of biological parameters that are associated with the scaffolder enzyme of the glucosome (i.e., PFKL) during the initiation of glucosome formation. Accordingly, we considered various scenarios that took into accounts of an intermolecular strength between PFKLs, an effective concentration of PFKL in a region of interest, an existence of second PFKL species due to alterations of PFKL properties at given environment, and the shape and flexibility of multivalent PFKL filaments. Then, we analyzed two outputs in details from our stochastic simulation: the frequency (%) of the formation of PFKL condensates and, when the condensates were formed, their size distribution. Collectively, our work here would provide mechanistic insights of how PFKL as a scaffold would initiate the formation of glucosome assemblies.Intermolecular interaction strengths on the formation of PFKL condensatesWe first investigated how different strengths of the PFKL-PFKL interaction affect the formation of PFKL condensates. To begin with, all \(512\) PFKLs were considered indistinguishable from each other, indicating that their mass, frictional force, interaction strength, and diffusion coefficients are identical. We then gradually increased intermolecular interaction strengths between each pair of PFKLs by modulating the depth of the potential well (i.e., \(\varepsilon\) values). The range of \(\varepsilon\) was determined to not only observe phase separation of PFKL but also allow us to quantify the formation of PFKL condensates in our stochastic simulation. The formation frequency of PFKL condensates was calculated by counting the number of occurrences that PFKL condensates appeared out of 300 independent runs of our simulations at each value of \(\varepsilon\) (Fig. 1A). The sizes of PFKL condensates were also obtained by counting the number of PFKLs forming each condensate at the end of the simulation time (i.e., \(1800 \; \text{s})\) (Fig. 1B).Figure 1Impacts of intermolecular interaction strengths on the formation of PFKL condensates. While we gradually increased enzyme-enzyme interaction strengths from \(\varepsilon =4.3\times {10}^{-23}\) to \(\varepsilon =5.2\times {10}^{-23}\), the formation frequency and the size distribution of PFKL condensates were measured and analyzed. (A) The frequencies of condensates formation and their standard errors at each interaction strength were plotted from 300 independent runs of simulations. (B) The size distributions of PFKL condensates were also shown at each interaction strength. Mean sizes of condensates (red lines) and standard errors of their sizes (green lines) at each interaction strength were graphed from 300 independent runs of simulations. (C) The size distributions of PFKL condensates at each interaction strength was fitted using multi-modal functions (i.e., double gaussian and exponential). The number of PFKL condensates (shown as the number of events) formed at a given size is graphed as green bars and the fitted size distributions are shown by black curves. Statistical analyses were performed using a two-way analysis of variance (ANOVA) along with Tukey’s multiple comparison tests. Statistical significance is defined as p < 0.05 with a 95% confidence interval while ‘ns’ refers to not significant. *p < 0.05, **p < 0.01, ***p < 0.001, ****p < 0.0001.As shown in Fig. 1A, we observed the formation of PFKL condensates in the range of \(\varepsilon\) from \(4.3\times {10}^{-23}\) to \(5.2\times {10}^{-23}\). The stronger enzyme-enzyme interaction was employed, the more frequent formation of PFKL condensates was observed. Briefly, when \(\varepsilon =4.3\times {10}^{-23}\) and \(\varepsilon =4.4\times {10}^{-23}\), PFKL condensates were rarely formed (\(<1\%\)). However, at \(\varepsilon =5.1\times {10}^{-23}\) and \(\varepsilon =5.2\times {10}^{-23},\) PFKLs were assembled into condensates in most runs of simulations (\(>99\%\)). In the latter case, we detected the coexistence of individual PFKLs and PFKL condensate(s) at the end of simulations. In addition, we compared sizes of PFKL condensates with a range of \(\varepsilon\) from \(4.5\times {10}^{-23}\) to \(5.2\times {10}^{-23}\) (Fig. 1B). We found that a proportion of the subpopulation of small-sized condensates that contained less than 100 PFKLs was reduced when the interaction strength between PFKLs was increased from \(\varepsilon =4.6\times {10}^{-23}\) to \(\varepsilon =5.2\times {10}^{-23}\) (Fig. 1B). However, the mean size of PFKL condensates increased as the interaction strength enhanced (red lines, Fig. 1B). Note that in the case of \(\varepsilon =4.5\times {10}^{-23}\), the mean size of PFKL condensates appeared to be biased due to the formation of PFKL condensates from 6 out of 300 runs of simulation. We also noticed that the size distributions of PFKL condensates appeared to be too broad at each \(\varepsilon\) value. By fitting their size distributions with multi-modal functions (i.e., double gaussian and exponential) (black lines, Fig. 1C), we found that as the interaction strength between PFKLs (\(\varepsilon\)) increased from \(4.7\times {10}^{-23}\) to \(5.2\times {10}^{-23}\), the mean size of a subpopulation of PFKL condensates that contained 400–512 PFKLs increased from 441.30 ± 5.42 to 495.73 ± 1.18, and the width of the subpopulation’s distribution became narrower from 32.49 ± 8.39 to 8.49 ± 1.18 (Figs. 1B,C). This result indicates that, as the interaction strength is increased within the range of our interest, condensates having 400–512 of PFKLs appear to be rapidly promoted by condensational growth that would narrow the size distribution, rather than collisional growth that would increase the width of condensates’ size distribution63,64. Overall, our stochastic simulation supports that the stronger enzyme-enzyme interaction is presented, the more frequent formation and the larger sizes of enzyme condensates are being formed.Correlation between an effective concentration of PFKL and its condensate formationWe then investigated how varying concentrations of PFKLs affect the size and frequency of condensate formation. We started our simulation with undistinguishable 512 PFKLs in the cubic domain. However, to increase an effective concentration of PFKL, we decreased the volume of the simulation cubic domain by every \(10\%\) from \(100\%\) (i.e., 1×) to \(60\%\) (i.e., 0.6×) of its default size. We evaluated this scenario at two different interaction strengths of \(\varepsilon =4.5\times {10}^{-23}\) (red in Fig. 2) and \(\varepsilon =4.7\times {10}^{-23}\) (green in Fig. 2). First, the frequencies of PFKL condensates formation were monitored at those \(\varepsilon\) values from different simulation volumes. From only ~ 2% at \(\varepsilon =4.5\times {10}^{-23}\) (i.e., 6 cases out of 300 simulations) or ~ 22% at \(\varepsilon =4.7\times {10}^{-23}\) (i.e., 67 cases out of 300 simulations), we observed the formation frequencies of PFKL condensates reaching ~ 100% (Fig. 2A). Second, we then compared condensate size distributions at different simulation volumes. Our data revealed that decreasing the simulation volume drastically increased the mean size of PFKL condensates (Fig. 2B), indicating that the higher effective concentration of PFKL at a region of interest would result in the larger size of a PFKL condensate. Therefore, the effective concentration of PFKLs at subcellular locations likely affects the size distribution of PFKL condensates in living cells.Figure 2Correlation between an effective concentration of PFKL and its condensate formation. To simulate the effects of various concentrations of PFKL at a given region of interest, we gradually reduced a simulation domain size from its default (\(1.0\times\)) to its 60% volume (\(0.6\times\)). The frequency of PFKL condensates formation (A) and their size distribution (B) were measured and analyzed at each simulation domain. Mean sizes of condensates (black lines) and standard errors (black error bars) are from 300 independent runs of simulations at each domain volume for two intermolecular interaction strengths of \(\varepsilon =4.5\times {10}^{-23}\) (red) and \(\varepsilon =4.7\times {10}^{-23}\) (green). Statistical analyses were performed using a two-way ANOVA along with Tukey’s multiple comparison tests. Statistical significance is defined as p < 0.05 with a 95% confidence interval. *p < 0.05, **p < 0.01, ***p < 0.001, ****p < 0.0001.Effect of second PFKL species with higher interaction strengthsIt is known that PFKL undergoes various posttranslational modifications. Specifically, acetylation at K689 is critical for the formation of glucosomes in living cells6. We hypothesized that the posttranslational modification of PFKL would affect its ability to form condensates by modulating its interaction strength. Therefore, we investigated if a second species of PFKL with different interaction strength would influence the formation of their condensates.We assumed that two different species of PFKLs would possess different levels of an intermolecular interaction strength (Fig. 3A). In this simulation, we assigned 128 out of 512 PFKLs as a second species (i.e., species 2) to have an ability to provide a stronger enzyme-enzyme interaction than the interaction strength the rest of PFKLs have (i.e., 384 PFKLs, species 1). Mathematically, we denoted the enzyme-enzyme interaction between species \(i\) and species \(j\) as \({\varepsilon }_{ij}\). Then, we considered three different scenarios, including a control, with various enzyme-enzyme interaction levels. First, we accounted for a control situation, as shown in Fig. 1, where the first and second species of PFKLs had the same enzyme-enzyme interaction strength. Relative levels of the interaction strengths between species \(i\) and species \(j\) were then expressed as \({\varepsilon }_{11}:{\varepsilon }_{12}:{\varepsilon }_{22}=1:1:1\) where \({\varepsilon }_{ij}={\varepsilon }_{0},\, i,j=\text{1,2}\). Next, we mimicked a situation that the second species of PFKLs were attracted to each other with a 20% stronger enzyme-enzyme interaction (i.e., \({\varepsilon }_{22}=1.2 {\varepsilon }_{0}\)) than a default interaction strength between the first species of PFKLs (i.e., \({\varepsilon }_{11}={\varepsilon }_{0}\)). We then assumed the two different interaction possibilities between species 1 and 2, creating two experimental sets of simulations: \({\varepsilon }_{11}:{\varepsilon }_{12}:{\varepsilon }_{22} = 1:1.1:1.2\) or \({\varepsilon }_{11}:{\varepsilon }_{12}:{\varepsilon }_{22}=1:1:1.2\). When we simulated these cases across three different default interaction strengths (i.e., \({\varepsilon }_{0}=4.6\times {10}^{-23}\), \(4.7\times {10}^{-23}\), and \(4.8\times {10}^{-23}\)), it becomes apparent that the presence of species 2 has a significant impact on the formation of PFKL condensates (Fig. 3A). At the same time, the size of PFKL condensates increased markedly as species 2 being introduced (Fig. S1). Collectively, we conclude that the addition of species 2 showing a slightly increased interaction strength, even in small amounts, enhances the capacity to initiate condensate formation.Figure 3Effects of existence of second PFKL species. We assumed that if PFKLs would form pre-organization prior to condensates formation, there could be at least two species of PFKLs where one might have a stronger interaction strength or display a heavier mass than the other might have. (A) First, we expressed the enzyme-enzyme interaction strength between species \(i\) and species \(j\) as \({\varepsilon }_{ij}, \,i,j=\text{1,2}.\) Then, three scenarios were considered when ratios of intermolecular interaction strengths between two species of PFKL were \({\varepsilon }_{11}:{\varepsilon }_{12}:{\varepsilon }_{22} = 1:1:1\) (blue), \({\varepsilon }_{11}:{\varepsilon }_{12}:{\varepsilon }_{22} = 1:1:1.2\) (red), or \({\varepsilon }_{11}:{\varepsilon }_{12}:{\varepsilon }_{22} = 1:1.1:1.2\) (green). Each scenario was tested in the case of species 1 possessing an enzyme-enzyme interaction strength of \({\varepsilon }_{11}=4.6\times {10}^{-23}\), \(4.7\times {10}^{-23}\), or \(4.8\times {10}^{-23}\). The frequency of PFKL condensates formation was plotted from 100 independent runs of simulations for each case. (B) Second, we simulated a case in which two species of PFKLs might experience different masses during condensate formation. Species 1 had a default value of the mass (\({m}_{0}\)) while species 2 had \(0.5\), \(1\), \(2\), \(4\), or \(8\) times of the default value of the mass. Both species had the same level of an enzyme-enzyme interaction strength, denoted as \(\varepsilon\). The frequencies of PFKL condensates formation and their standard errors were plotted from 300 independent runs of simulations for each case at \(\varepsilon =4.8\times {10}^{-23}\) (red) or \(\varepsilon =4.9\times {10}^{-23}\) (green).Effects of second PFKL species as being part of a pre-organizationMeanwhile, metabolic enzymes have been discovered to form various spatially resolved structures in living cells. In facts, mesoscale filaments, rods, rings, sheets, lattices, tubes as well as liquid-like condensates have been observed under fluorescence microscopy and/or electron microscopy43,65,66. It seems logical to speculate that such mesoscale structures formed by metabolic enzymes might be constructed from nanoscale pre-organizations. Accordingly, we hypothesized that, if PFKL was organized into any form of pre-organization, each PFKL unit in a pre-organization might experience like having a heavier mass than ones that freely diffusing PFKLs would do.We introduced a heavier mass species to distinguish PFKLs in pre-organization from free PFKLs. In this simulation, we considered 384 PFKLs having a default mass (\({m}_{0})\) and 128 PFKLs with a mass that was equal to \(0.5{ m}_{0}\), \({m}_{0}\), \(2{m}_{0}\), \(4{m}_{0}\), or \(8{m}_{0}\). Other than the mass, both species of PFKLs were assumed to have the same physical properties. We then evaluated each scenario at \(\varepsilon =4.8\times {10}^{-23}\) or \(\varepsilon =4.9\times {10}^{-23}\). Note that the two \(\varepsilon\) values were chosen so that both free PFKLs and PFKL condensates would coexist during simulations. When two species of PFKLs had the same default mass (\({m}_{0})\), the frequency of PFKL condensates formation were \(42.7\%\) and \(70.0\%\) at \(\varepsilon =4.8\times {10}^{-23}\) and \(\varepsilon =4.9\times {10}^{-23}\), respectively (Figs. 1A and 3B). However, as we varied the mass of species 2 PFKL, we did not see a significant change in the frequency of PFKL condensates formation (Fig. 3B). Therefore, we conclude that the frequency of PFKL condensates formation may not significantly depend on additional PFKL species displaying different masses. Note that the size distributions of PFKL condensates did not change significantly either as we varied the mass of species 2 PFKL (Fig. S2).Multivalent interactions and structural variability of PFKL filaments in condensate formationMultivalent constituents such as polymer and filaments have been known to organize biomolecular condensates1,43,67. In fact, PFKLs have been demonstrated to form nanoscale filaments in vitro44,68. Additionally, PFKL’s diffusion within glucosomes slows down by more than an order of magnitude compared to its diffusion outside glucosomes6,42. We hypothesized that PFKL filaments might be critical components as multivalent scaffolds to initiate the assembly of PFKL condensates and thus to structure glucosomes.To test the hypothesis, we introduced PFKL filaments with two variables, the flexibility and the length of the filaments, in our stochastic simulations. We first defined that species 1 was a free PFKL and species 2 was an oligomeric PFKL filament (Fig. 4). A total of four PFKL filaments of the same lengths were used in each simulation. These filaments were composed of multiple PFKLs (i.e., 3, 4, 5, or 6), defining their lengths. In all scenarios, the total number of PFKLs, regardless of their participation in filaments, remained at 512. We then implemented various lengths of PFKL filaments in two ways (Fig. 4). First, ‘elastic’ PFKL filaments were modeled with PFKLs connected via freely-rotating interaction points. Second, ‘rigid’ PFKL filaments were introduced with PFKLs connected at fixed angles to maintain a \(180^\circ\) alignment. To examine the effects of PFKL filaments at varying interaction strengths, our simulations in each scenario were performed at \(\varepsilon =4.5\times {10}^{-23}\) and \(\varepsilon =3.3\times {10}^{-23}\). Note that \(\varepsilon =4.5\times {10}^{-23}\) was determined as the threshold for forming PFKL condensates when only free PFKLs were present (Fig. 1B) while the interaction strength, \(\varepsilon =3.3\times {10}^{-23}\), was selected in our simulation because this value is significantly below the threshold (i.e., 26.7% lower than \(\varepsilon =4.5\times {10}^{-23}\)).Figure 4Contribution of PFKL filaments to the formation of condensates and their sizes. In this simulation, we assumed two species of PFKLs: species 1 was a freely diffusing individual PFKL whereas species 2 was an oligomerized PFKL filament consisting of 3–6 PFKLs. Elastic (orange and red) or rigid (blue and green) filaments having different lengths were simulated at the enzyme-enzyme interactions of \(\varepsilon =3.3\times {10}^{-23}\) (orange and blue) or \(\varepsilon =4.5\times {10}^{-23}\) (red and green). 300 independent runs of simulations were performed at each case. (A) The frequency (%) of PFKL condensates formation was graphed as varying the flexibility and the length of filaments. (B,C) The size distributions of observed PFKL condensates were displayed at each enzyme-enzyme interaction strength for each filament length. (D) The size distributions of PFKL condensates were also fitted using multi-modal functions. The numbers of PFKL condensates formed at a given size are graphed as bars and the fitted size distributions are shown by black curves. Statistical analyses were performed using a two-way ANOVA along with Tukey’s multiple comparison tests. Statistical significance is defined as p < 0.05 with a 95% confidence interval. *p < 0.05, ****p < 0.0001.Upon performing 300 runs of each simulation, we observed positive correlations between the length of PFKL filaments and the formation frequency of PFKL condensates (Fig. 4A). At an interaction strength of \(\varepsilon =4.5\times {10}^{-23}\), free PFKLs barely organized into condensates in the absence of a filamentous structure (Fig. 1). However, we observed that the introduction of a higher multivalent filament species, regardless of its flexibility, strongly facilitated condensate formation (red and green, Fig. 4A). Even at a significantly weaker enzyme-enzyme interaction strength of \(\varepsilon =3.3\times {10}^{-23}\), condensate formations were apparent. In fact, elastic PFKL filaments were more effective in recruiting free PFKLs to initiate condensate formation compared to rigid PFKL filaments (orange vs. blue, Figs. 4A, and S3–S4). The more flexible and longer filaments exist, the more effective formation of PFKL condensates is observed. In conclusion, the multivalency provided by filament structures significantly enhances the formation of PFKL condensates even at a significantly lower interaction strength than the threshold of the PFKL condensate formation.Impact of PFKL filaments on condensate sizes for functional implicationPreviously, we have demonstrated that glucosome condensates form in various sizes at a subcellular level6,8,42. Importantly, we have demonstrated that their metabolic functions are directly associated with their sizes in living cells6,8,42. Given the significance of PFKL filaments on the condensate formation as shown in Figs. 4A and S5, we analyzed size distributions across scenarios to address if the condensate sizes would be influenced by the PFKL filaments’ efficacy of initiating condensate formation within our simulation time.First, we investigated the effect of elastic filaments in condensate size distribution. At an interaction strength of \(\varepsilon =3.3\times {10}^{-23}\), PFKL condensates were generally formed in larger sizes with elastic filaments than with rigid ones (orange vs. blue, Fig. 4B). It seems that elastic PFKL filaments may effectively interact with other PFKL molecules, facilitating the formation of early condensates and their sustained growth during the 30-min simulation. Additionally, when the length of PFKL filaments was increased, we consistently observed the formation of larger condensates in size. Interestingly, with elastic filaments, two distinct subpopulations of condensates were monitored when the elastic filaments were composed of 4, 5, or 6 PFKLs, thereby requiring a double Gaussian fit for size distribution analysis (highlighted in orange in Fig. 4D). As the length of elastic filament increased from 4 to 5 PFKLs, the average size of both the subpopulations of condensates slightly increased by 18.1% and 11.1% respectively, with the quantity of condensates nearly doubling. However, an obvious inflection point was observed when the length of elastic filaments increased from 5 to 6 PFKLs because the mean sizes of the two subpopulations of condensates increased by 115.8% and 40.0%, respectively. Yet, the frequencies of both subpopulations in the case with elastic filaments composed of 6 PFKLs remained consistent with those observed with elastic filaments consisting of 5 PFKLs. Meanwhile, with elastic filaments at an interaction strength of \(\varepsilon =4.5\times {10}^{-23}\), the size distributions showed a dominant subpopulation of large condensates (over 400 in size) but did not show a significant shift in size with increasing filament lengths (red, Fig. 4D). In conclusion, we show that the length of elastic filaments determines the size of condensates, and even one additional PFKL unit in growing elastic filaments would be capable of shifting size distribution of condensates significantly if the number of PFKL unit in elastic filaments reaches an inflection point.Second, we investigated the effect of rigid filaments in condensate size distribution. In contrast to the case of elastic filaments, at an interaction strength of \(\varepsilon =3.3\times {10}^{-23}\), the size distributions of condensates from rigid filaments with 4 or 5 PFKLs displayed a unimodal pattern. Interestingly, the mean sizes of these condensates were comparable to those of the smaller-sized condensates formed from elastic filaments with an equivalent number of PFKLs (blue, Fig. 4D). However, introducing a rigid filament with 6 PFKLs resulted in a bimodal distribution. Unlike with elastic filaments, the smaller condensate population from these rigid filaments showed no increased size whereas the larger condensate population exhibited a 43.2% increase in its mean size compared to that formed from shorter rigid filaments. Similar to the case of elastic filaments, at \(\varepsilon =4.5\times {10}^{-23}\) rigid filaments did not show a significant shift in size with increasing filament lengths (green, Fig. 4D). In short, these findings suggest that rigid filaments, relative to elastic filaments, do not significantly influence the size of condensates.
Cross interaction between filaments in PFKL condensate formationTo understand whether cross-interactions between filaments are required to form condensates, we measured whether a single filament could form PFKL condensates. When the interaction strength was set as \(\varepsilon =4.5\times {10}^{-23}\), many condensates were composed of 1 filament with various number of PFKLs (Table 2). The efficiency of condensate formation with a single filament got higher as the length of filament got longer. Even with the interaction strength at \(\varepsilon =3.3\times {10}^{-23}\), which was well below the threshold of free PFKL’s condensation, we still observed a condensate being formed with a single elastic filament of 5 or 6 PFKLs (Table 3). Notably, under strong interaction settings or in the presence of longer filaments, all four filaments that we started with the simulation mostly ended up partitioning into condensates, especially for those containing larger than 400 PFKLs.
Table 2 Partition numbers of PFKL pre-organizations included in each PFKL condensate at \(\varepsilon =4.5\times {10}^{-23}\).Table 3 Partition numbers of PFKL pre-organizations included in each PFKL condensate at \(\varepsilon =3.3\times {10}^{-23}\).
Mechanistic insights into condensate formation of human liver-type phosphofructokinase by stochastic modeling approaches
