ABSTRACT
The concept of critical concentration is a central idea in understanding the behaviors of microtubules (MTs) and other steady-state (energy-utilizing, non-equilibrium) cytoskeletal polymers. Classically, the critical concentration (CC) is the concentration of subunits necessary to obtain polymer. However, the classical theory used to explain and predict CC is based on equilibrium polymers and fails to account for dynamic instability (DI). It has been unclear how the classical theory should be adjusted to incorporate DI or how the behavior of an individual dynamically unstable filament relates to that of its population. To address these questions, we used previously established simulations to follow at multiple scales the behavior of systems of computationally modeled dynamic microtubules. We show that polymers such as microtubules that exhibit dynamic instability have not one but at least two critical concentrations: one above which growth phases of individual filaments can occur transiently, and another above which the population’s overall polymer mass will increase persistently. We propose that whether a steady-state polymer behaves like microtubules (displays dynamic instability) or like actin (can be modeled as an equilibrium polymer) depends on how far apart these critical concentrations are. This revised framework helps to explain and unify a wide range of experimental observations.
SIGNIFICANCE STATEMENT Traditionally, biological polymers are thought to have one “critical concentration” (CC) where polymer assembly commences. It has been unclear how this framework should be adjusted to account for dynamic instability (prolonged periods of growth and shortening with random transitions). Using computational models, we show that microtubules have two critical concentrations: a lower CC where individual dynamically unstable microtubules can grow transiently (CCIndGrow), and an upper CC where a population of such filaments grows persistently (CCPopGrow). Our results show that most experiments thought to measure “the CC” actually measure CCPopGrow. “Normal” dynamic instability (where individual filaments depolymerize back to the seed) happens at subunit concentrations below what is traditionally thought to be the lower lower limit for polymer appearance.
Introduction
The concept of critical concentration (CC) is fundamental to experimental studies of biological polymers, including microtubules (MTs) and actin, because it is used to determine the amount of subunit needed to obtain polymer. In the standard framework for predicting the behavior of biological polymers, there is one critical concentration at which polymer assembly commences (e.g., (1, 2)). However, this framework fails to account for the dynamic instability (DI) behavior of microtubules. The purpose of the work presented here is to adjust the classical understanding of critical concentration to incorporate dynamic instability. We find that DI polymers like microtubules have not one critical concentration, but at least two: a lower critical concentration at which individual microtubules can grow transiently, and a higher critical concentration at which a population of microtubules will grow persistently. Most measurements of critical concentration detect the higher value, the critical concentration for persistent population growth. What might be considered “normal” dynamic instability (i.e., dynamic instability where the MTs achieve a steady-state average length) happens at [free tubulin] between these two critical concentration values.
Traditional understanding of Critical Concentration in biological polymers
Two key observations led to the standard understanding of critical concentration and were originally made in very early studies of actin. First, no significant polymer was obtained until the total concentration of monomers reached an empirically defined concentration termed the “critical concentration” (CC) (Figure 1A, label Q1), above which polymer formed (3). Second, once the concentration of total monomers (subunits) reached this CC for polymer assembly, any additional subunits beyond this CC value were converted into polymer. The end result was that after polymerization reached its steady-state value, the concentration of subunits left in solution (Figure 1A, label Q2) was approximately equivalent to the observed critical concentration for polymer assembly (3–5).
These experimental observations were given a theoretical framework by Oosawa and colleagues, who explained the behavior of actin by developing a theory for the equilibrium assembly behavior of helical polymers (5, 6). This theory was extended to tubulin by Johnson and Borisy (4). The idea that polymer assembly commences at the critical concentration is now used routinely to design and interpret experiments involving cytoskeletal polymers (e.g., (2, 7–10)), and it is a standard topic for cell biology textbooks (e.g., (1, 11)). Over time, a set of experimental measurements and definitions of critical concentration, all generally treated as equivalent, have emerged (Table 1).
On the surface, it seems reasonable to apply this framework to understanding polymers like microtubules: it is broadly consistent with many experimental results (12), and it is founded on fundamental concepts of biochemical equilibria (e.g., see footnotes to Table 1). However, in the case of microtubules and other polymers that exhibit dynamic instability (stochastic switching between phases of growth and shortening, Figure 1C), it has been unclear how the critical concentration definitions in Table 1 relate to the behaviors of individual filaments and populations. There is also a deeper problem with this traditional approach: the theoretical foundation of the framework outlined above assumes that the polymers being studied are equilibrium polymers, i.e., the analysis assumes that the polymer system is at thermodynamic equilibrium. At equilibrium, no energy enters or leaves the system. In contrast, microtubules are said to be steady-state polymers because they require a consistent input of energy (in the form of a pool of GTP) to maintain polymerization. Steady-state polymers like microtubules will be (mostly) disassembled at true thermodynamic equilibrium because the nucleotides in the system will be entirely hydrolyzed.
The issue of using equilibrium polymer theory to describe the behavior of steady-state polymers has long been recognized as being potentially problematic (3, 12–13). Nevertheless, it has been unclear how the impact of nucleotide hydrolysis should be incorporated into the outline above or how dynamic instability relates to these classical definitions of critical concentration. Hill and colleagues investigated problems related to these questions in the mid-1980s. The results of their work suggest that equilibrium models are limited in their ability to predict the behavior of steady-state polymer systems. In particular, they found evidence for two distinct critical concentrations in their microtubule simulations (e.g., (14), summarized in (15)). However, the existence of two critical concentrations in DI polymers has not been incorporated into common understanding, perhaps because Hill’s early pioneering work did not clarify for readers the relationship between the behaviors of individual filaments and those of their populations. Thus, the equilibrium-based models remain the primary framework in use for understanding biological polymer systems.
Empirical evidence that the problem of critical concentration deserves attention comes from the observation that experimentally reported CC values for mammalian brain tubulin range from less than 3 µM to more than 20 µM (e.g. (2, 9, 16–18)). One potential explanation is that these differences can be attributed to variation in experimental approach, buffer conditions, or tubulin quality. However, another possibility is that at least some of the variation results from inaccurately applying equilibrium theory to a steady-state polymer system that exhibits dynamic instability. Proper application of critical concentration measurements to the design and interpretation of future experiments requires consensus on the definition of critical concentration for microtubules and similar polymers.
Computer simulations as an approach to addressing these questions
To investigate the concept of critical concentration in dynamically unstable polymers, we have used two different stochastic computational models that both simulate systems of dynamic microtubules, but do so at different levels of detail (19, 20). Computational models are ideal for addressing this type of problem because the biochemistry of the reactions can be explicitly controlled. Furthermore, “experiments” can be performed quickly and easily under conditions where it is possible to simultaneously follow the behavior of the system at all relevant scales: addition/loss of individual subunits, dynamic instability of individual filaments, and any changes in polymer mass of the population of filaments. In comparison, the necessary multi-scale analysis of microtubule systems has been difficult to achieve experimentally because experiments have thus far been limited technically to measurements at one (or at most two) of these scales at a time.
In our simulations (19, 20), the governing rules and conditions are those that would be set by the experimenter or by intrinsic properties of the biological system. More specifically, the user defines the kinetic behavior of the subunits (i.e., rates of bond formation/rupture, GTP hydrolysis), the concentration of tubulin subunits in the system, whether the system is closed (competing) or open (non-competing), and the volume. Because microtubules in cells and in many in vitro experiments grow from stable seeds (nucleation sites), our simulations assume that one end is fixed (as would be the case for growth from centrosomes), and that all addition and loss occurs from the free end. For simplicity, the number of stable MT seeds is usually constant, but in some simulations we explore the impact of changing the number of stable seeds. Typical experimental results (DI parameters, concentrations of free and polymerized tubulin) are emergent properties of the system of biochemical reactions, just as they would be in a physical experiment.
Summary of Conclusions
Using these systems of simulated microtubules, we show that classical interpretations of experiments such as those shown in Figure 1 can be very misleading in terms of understanding the behavior of individual MTs. In particular, we find that dynamically unstable polymers like microtubules have not one but at least two experimentally distinguishable critical concentrations: a lower CC where growth phases of individual filaments can occur transiently, and a higher CC where growth of the population’s polymer mass will occur steadily (even while individual filaments in this population potentially still exhibit dynamic instability). To distinguish these two CC values, we call them CCIndGrow and CCPopGrow, respectively. One key conclusion is that most experiments intended to measure the CC actually measure CCPopGrow. A related conclusion is that “normal” microtubule dynamic instability (where MTs grow and depolymerize back to the seed) is limited to concentrations below what has classically been considered “the” CC needed for polymer assembly. The improved understanding of microtubule dynamics that results from these studies helps explain a range of in vitro microtubule experiments and provides a more solid foundation for understanding how in vitro microtubule behavior relates to that observed in vivo. While our studies focus on microtubules, we suggest that our critical concentration definitions and interpretations will apply to any steady-state polymer and are especially significant to those that exhibit dynamic instability.
Results and Discussion
Computational Models
In this work, we used both a “simplified” model of MT dynamics, in which MTs are modeled as simple linear polymers (19), and a “detailed” model, where microtubules are composed of 13 protofilaments, with individual lateral and longitudinal bonds between subunits (tubulin dimers) modeled explicitly (20) (Figure 2A-B). Subunit addition/loss and GTP hydrolysis (both models) and lateral bond formation/breaking (detailed model only) are stochastic events that occur according to user-defined rate constants. As noted above, the dynamic instability parameters and the concentrations of free and polymerized tubulin subunits (abbreviated as [free Tu] and [MT polymer] respectively) at steady state are emergent properties of the systems, just as they would be in physical experiments. Both simulations spontaneously undergo the full range of dynamic instability behaviors (including rescue), and they can simulate systems of dynamic microtubules (either competing, with constant [total tubulin], or non-competing, with constant [free tubulin]) for hours of simulated time (19, 20). These attributes make the simulations ideal for the goal of understanding the behavior of systems of dynamic MTs and how this behavior relates to that of individual filaments.
The parameters of the detailed model were previously tuned to approximate the behavior of MTs in vitro (20); the simplified model parameters used here are modified from those of (19), which were originally tuned to match in vivo MT behavior. These two models produce dynamic instability behavior that is quantitatively different (see Tables S1-S2 for DI parameters observed from these simulations, and Supplementary Information for more discussion of the simulations and their input parameters). Thus, while it is expected that any specific values for critical concentration(s) extracted from these two simulations will be different, we can use these differences to help address the questions of which conclusions are general and which (if any) might be specific to certain parameter sets or polymer types.
Initial Considerations
To begin investigating the concept of critical concentration as it applies to microtubules, we first asked whether the definitions outlined in Table 1 are valid when studying microtubules, and if so, whether they are equivalent to each other.
The easiest of the various CC definitions to consider is CCKD: although the idea that CCKD = koff/kon = KD is frequently stated, it also is well-recognized that this relationship cannot be applied in a straightforward way to populations of dynamic microtubules (1). More specifically, experimentally observed critical concentrations for systems of dynamic MTs (however measured) cannot be equated to simple koff/kon = KD values because the GTP and GDP forms of tubulin have significantly different kon and koff values and therefore significantly different KD values. For example, the critical concentration for GMPCPP (GTP-like) tubulin has been reported to be less than 1 µM (21), while that for GDP tubulin is very high, perhaps immeasurably so (12).
Exactly how the measured CC value for a system of dynamic microtubules relates to the KD values for GTP and GDP tubulin is not yet known. However, intuition suggests that it must lie between the respective values for GTP and GDP tubulin (12). Consistent with this idea, experimentally reported values for tubulin CC typically lie between ~3 and ~20 µM (9, 16–18). Note that while the idea that CC = koff/kon = KD cannot apply in a simple way to a system of dynamic microtubules, it does apply to GDP tubulin (when polymerized with certain drugs) or to systems containing only non-hydrolyzable (e.g., GTP-γS) or slowly-hydrolyzable (e.g., GMPCPP) analogs because in the absence of hydrolysis tubulin assembly is an equilibrium phenomenon (8, 22).
The invalidity of the idea that CC = KD raises questions about the other definitions of CC, given that they are all generally treated as equivalent. First we will investigate in more detail the two most familiar definitions for CC (see also Table 1):
CCPolAssem = [total subunit] needed for polymers to assemble.
CCSubSoln = [subunits left in solution] once steady-state polymer formation has been achieved.
In Table 1 and the discussion below, we will use the terms Q1, Q2, etc. to refer to specific experimentally measurable quantities (i.e., values obtained through experimental approaches as indicated in the figures), and the terms CCPolAssem, CCSubSoln, etc. to refer to theoretical values (concepts) that may or may not correspond to particular experimentally measurable quantities.
Standard experimental approaches for measuring critical concentration do not yield the results predicted from traditional understanding
As discussed above, the expectation from equilibrium polymer theory is that CCPolAssem = CCSubSoln, and that the value of CCPolAssem = CCSubSoln can be experimentally measured by determining Q1 or Q2 in an experiment such as those portrayed in Figure 1A. In other words, the expectation is that Q1 ≈ Q2, and that these experimentally obtained values provide equivalent ways to measure “the” critical concentration for polymer assembly.
We first asked whether CCPolAssem and CCSubSoln are equivalent for microtubules by performing simulations of “closed” systems where individual MTs compete for a limited pool of tubulin subunits ([total tubulin] is constant). This situation is analogous to an experiment in a test tube in which microtubules grow from pre-formed MT seeds, and both [polymer] and [free subunit] are measured at steady state.* At first glance, the behavior of the systems of simulated microtubules seemed consistent with that expected from common understanding (Figure 1A): significant polymer assembly was first observed at [total tubulin] ≈ Q1, and Q1 ≈ Q2 (Figure 3AB).
However, closer examination of these data showed that it is difficult to determine exactly where Q1 and Q2 are. In particular, small but non-zero amounts of polymer exist at [total tubulin] below reasonable estimates for Q1 based on extrapolating the [polymer] data back to [polymer] = 0 (Figure S1C-F). Consistent with this observation, when we examined the behavior of the individual MTs in these simulations, we saw MTs growing and exhibiting dynamic instability at total tubulin concentrations below Q1 (Figure 3C-D; compare to Figure 3A-B). The observation that MTs exist at [total tubulin] below “the” CC as determined by Q1 is contrary to the traditional understanding of critical concentration.
In addition, it is often assumed that the concentration of unpolymerized tubulin left in solution at steady state is independent of [total tubulin] once a threshold [total tubulin] (i.e., the CC) has been reached (e.g., (23)). However, it can be seen in Figures 3A-B and S1E-F that steady-state [free tubulin] is not constant with respect to [total tubulin], and instead approaches a plateau represented by Q2 ≈ CCSubSoln. In other words, CCSubSoln is the plateau approached by the [subunits left in solution], not the observed value of [subunits left in solution] at any particular [total subunit].
A related issue is that depending on how the Q1 extrapolation is performed, the values of Q1 and Q2 might appear different. However, if we assume that Q2 is the concentration that [free tubulin] approaches as [total tubulin] increases, and Q1 is the horizontal intercept of a line (with slope 1) that [polymer] approaches as [total tubulin] increases, then Q1 ≈ Q2 (Figure 3AB), as expected. Regardless, it is evident that polymers exhibiting dynamic instability appear at [total tubulin] below Q1 ≈ Q2 (Figures 3, S1).
The sum of these data indicates that two of the major and commonly accepted predictions of equilibrium polymer theory are invalid when applied to systems of dynamic microtubules:
Instead of both Q1 and Q2 providing an experimental measure of the concentration of subunits needed for polymer assemble (CCPolAssem), neither does, since dynamically unstable MTs can be observed at concentrations below Q1 and Q2.
Instead of all definitions of critical concentration being equivalent, CCSubSoln ≠ CCPolAssem because MTs can be observed at concentrations lower than CCSubSoln as measured by Q2.
The observations thus far raise two questions: (i) If CCSubSoln ≠ CCPolAssem, what (if anything) is the significance of Q2 ≈ CCSubSoln for microtubule behavior? (ii) Is there an actual CCPolAssem, i.e., a specific concentration of total subunits at which MT polymers begin to assemble?
A critical concentration for persistent growth of MT populations (CCPopGrow)
We investigated the significance of Q2, i.e., the asymptote approached by [free tubulin] at steady state, by using our simulations to examine the amount of polymer and the mean MT length at steady state as functions of the concentration of free tubulin. For these particular studies, we fixed [free tubulin] at the indicated values instead of allowing polymer growth to deplete the free tubulin. This set of conditions is analogous to a laboratory experiment involving MTs polymerizing from stable seeds in a constantly replenishing pool of free tubulin at known concentration, such as might exist in a flow cell. Figure 4 shows the results of the simulation experiments:
At low concentrations of free tubulin, there is little if any detectable MT assembly, as one might expect (Figures 4A-B, S2A-B).
At intermediate free tubulin concentrations, both the average MT length and the concentration of MT polymer within a population reach steady-state values that increase with free tubulin but are constant with time (Figures 4C-D, S2A-B; note that these graphs have two vertical axes). In other words, the systems reach a polymer mass steady state. Individual microtubules in these systems exhibit what might be called “typical” dynamic instability: they undergo periods of growth and shortening with approximately random transitions (called “catastrophe” and “rescue,” see Figure 1C), but they eventually and repeatedly depolymerize back to the stable GTP-seed (compare Figure 4A-B to Figures 4C-D and S2A-B).
At still higher free tubulin concentrations, the populations of dynamic MTs undergo a major change in behavior: they begin to grow persistently. More specifically, when [free tubulin] is above a threshold value, there is no steady state at which the concentration of polymerized tubulin is constant over time (i.e., there is no polymer mass steady state). Instead, the system of MTs arrives at a different type of steady state where polymer mass increases at a constant rate (Figures 4C-D, S2A-B). In other words, the system of MTs reaches a polymer growth steady state. Individual MTs within these populations still exhibit dynamic instability (except perhaps at very high [free tubulin]), but they exhibit net growth over sufficient time (compare Figure 4A-B to Figures 4C-D and S2A-B).
The threshold at which this behavior change occurs (Q5) can be identified from a plot of the rate of change in the steady-state polymer mass as a function of [free tubulin] (Figure 4C-D). Examination of this plot indicates that at concentrations of free tubulin below Q5, the population will eventually arrive at a polymer mass steady state (i.e., the rate of change in polymer mass with time ≈ zero). In contrast, at [free tubulin] above Q5, the population will increase in mass at a constant rate that increases with the concentration of free tubulin.
Consistent with the idea that there is a [free tubulin] at which MTs begin to exhibit net growth, Dogterom and colleagues previously predicted the existence of a critical concentration at which MTs transition from exhibiting “bounded growth” to “unbounded growth”† (36, 37). As discussed more below, the critical concentration as predicted by Dogterom’s equation evaluated with our DI measurements (Tables S1-S2) matches Q5 (compare + symbols to circles in Figure 4C-D).
A more experimentally tractable way to identify this threshold concentration for persistent growth of MT populations is to analyze the behavior of many individual MTs within a population according to the diffusion-drift paradigm of Borisy and colleagues (25, 26). As can be seen in Figure 4E-F, a population of MTs exhibits zero drift at free tubulin concentrations below Q6 but exhibits positive drift at [free tubulin] above Q6. As one might intuitively predict, Q5 ≈ Q6.
On the basis of these observations, we conclude that microtubule populations undergo a persistent increase in polymer mass at free tubulin concentrations above the value identified by Q5 ≈ Q6. As averaged over long periods of time or over many individuals, the MTs in these persistently growing populations undergo net growth and positive drift, all while still experiencing dynamic instability (except perhaps at the highest tubulin concentrations). In contrast, at [free tubulin] below Q5 ≈ Q6, MT populations will (if given sufficient time) arrive at a steady-state polymer mass at which individual MTs will have a steady-state average length and experience zero net drift.
Significantly, Q5 ≈ Q6 lies at approximately the value of Q1 ≈ Q2, i.e., ~2.8 µM for the simplified model, and ~11.8 µM for the detailed model. This observation means that steady-state [free tubulin] in competing systems asymptotically approaches the same [free tubulin] at which microtubules begin to exhibit net growth (positive drift) in non-competing systems (compare Q2 in Figure 3A-B to Q5 ≈ Q6 in Figure 4C-F). Taken together, these observations show that the experimentally derived quantity Q1 ≈ Q2 ≈ Q5 ≈ Q6 does not yield the critical concentration for polymer assembly (CCPolAssem) as expected from traditional understanding. Instead, these measurements yield a different CC: the CC for persistent population growth (CCPopGrow).
As noted above, Dogterom and colleagues identified a critical concentration for the transition from bounded to unbounded growth,✝where bounded and unbounded growth are characterized by the following equation (36):
Below the [free tubulin] at which the transition occurs, the average MT length reaches a finite steady-state value where the rate of change in average length is JDogterom = 0 (i.e., population growth is “bounded”). Above this concentration, the average MT length increases with time at a rate of JDogterom > 0 (i.e., population growth is “unbounded”).
To determine how the critical concentration as predicted by the JDogterom equation relates to the critical concentrations discussed above, we measured the DI parameters in the simulations across a range of [free tubulin] (Tables S1, S2), input the DI data into this equation, and compared the results to those obtained from directly measuring changes in average MT length (Figure 4C-D). Strikingly, the rates of change in average length as calculated from the DI measurements of individual MTs using the JDogterom equation are very similar to the rates calculated from the net change in average length of the population of filaments (Figure 4C-D, compare + symbols to circles). Notably, the JDogterom equation outputs become positive at approximately the same [free tubulin] as Q5. This means that the critical concentration for unbounded growth identified by Dogterom, et al. (36, 37) agrees with the CC measured by Q1, Q2, Q5 and Q6 seen in Figures 3–4.
Thus, all of the approaches to measuring critical concentration discussed thus far yield the critical concentration for persistent population growth (CCPopGrow). This conclusion leaves us with two unresolved questions: (i) What is the significance of the remaining experimental CC measurements Q3 and Q4 (Table 1, Figures 5–6)? (ii) Is there a CC at which DI polymers appear?
A critical concentration for transient elongation (growth) of individual filaments (CCelongation = CCIndGrow)
Q3 has previously been used as a measure of the so-called “critical concentration for elongation” (CCe) (27). According to standard models, CCe is the free subunit concentration where the rate of subunit addition to an individual filament exactly matches the rate of subunit loss from that individual filament, meaning that individual filaments should grow at subunit concentrations above CCe (see Table 1 and its footnotes). Thus, it is reasonable to predict that polymer assembly will commence at CCe, implying that CCe = CCPolAssem.
To determine Q3 in our simulations, we used the standard approach for MTs outlined in Table 1 (27): we plotted the growth velocity (Vg) of individual filaments as observed during the growth phase of dynamic instability as a function of [free tubulin], and extrapolated that linear relationship back to Vg = 0. We obtained Q3 (and thus approximate CCe) values of ~0.65 µM for the simplified model and ~4.2 µM for the detailed model (Figure 5A-B). Comparing these CCe values to the data in Figures 3 and 4 shows that CCe as measured by Q3 (Figure 5A-B) is well below CCPopGrow as measured by any of the other approaches (Q1 ≈ Q2 ≈ Q5 ≈ Q6). This observation demonstrates that Q3 provides information about MT behavior not provided by the other measurements.
However, the prediction that CCe = CCPolAssem fails. Contrary to traditional expectation, there is no total or free subunit concentration at which polymer assembly commences abruptly. Instead, the amount of polymer initially increases in a slow and nonlinear way with respect to [free tubulin], increasing more rapidly only as [free tubulin] approaches CCPopGrow (Figures 3–4, S1-S2). The same conclusion is reached whether examining polymer mass (Figures 3A-B, S1E-F), average MT length (Figure S2A-D), or maximal MT length (Figure S2C-D). These observations mean that microtubules do not have what might be classically described as a critical concentration for polymer appearance (the theoretical CCPolAssem discussed earlier).
Given this information, what is the significance of CCe? According to the approach outlined above, CCe is the [free tubulin] value above which the growth velocity first becomes positive. In considering this question for DI polymers, it is important to remember that CCe, as estimated by Q3, is determined from measurements of the growth velocity of individuals during the growth phase of dynamic instability. Thus, CCe provides the minimum concentration of tubulin needed for individual filaments to grow transiently (i.e., to extend during the growth phase of dynamic instability).
In contrast, as discussed above, CCPopGrow is determined from population-level measurements that do not separate growth and shortening behaviors (e.g., the rate of change in average length in Figure 4C-D is a net rate of change over both growth and shortening phases). CCPopGrow provides the minimum [free tubulin] necessary for the population of MTs (i.e., the bulk polymer) to grow persistently. When [free tubulin] is above CCPopGrow, individuals may still display DI, but over time they exhibit net growth (i.e., growth outweighs shortening; on average, the MTs continually get longer with time). When [free tubulin] is below CCPopGrow, individuals repeatedly depolymerize back to the seed (compare length history plots above and below CCPopGrow in Figure 4A-B). Consistent with the identification of the upper CC as CCPopulationGrowth (CCPopGrow), we suggest referring to CCe as CCIndividualGrowth (CCIndGrow).
In considering the significance of CCe, it is important to recognize that Q3 provides only an approximate measure of CCe. This is because at least two of the assumptions used to measure CCe via Q3 (see Table 1, footnote d) are likely to be invalid, at least for mammalian brain microtubules. First, measurement of CCe via Q3 assumes that the velocity of growth is Vg = kon_GTP [GTP-tubulin] – koff_GTP, which implies that the off rate is independent of [free tubulin]. However, recent in vitro experiments have shown that the per protofilament off rate for tubulin detaching from a growing MT does depend on the tubulin concentration (20, 28, 29). This dependence of the off rate on [free tubulin] likely results from concentration-dependent differences in tip structure (20, 28, 29). Second, work with our simulations indicates that the assumption in classical CCe measurements that only GTP subunits are exposed at the tip is likely to be invalid: GDP tubulin subunits frequently but transiently occupy the terminal position of growing (proto)filaments in both the simplified and detailed models (19, 20, 30). These observations mean that Q3 is at best an approximation (perhaps a bad one) of CCe. Similar arguments hold for why Q3 provides at best only an approximate measure of KD_GTP (see Table 1, footnote d).
Appearance of MTs on stable seeds mimics a nucleation-like process without necessarily involving one
Two experimental observations have previously led to the idea that growth of MTs from stable nucleating structures (MT seeds or centrosomes) involves a nucleation-like step: (i) the fraction of stable nucleating structures occupied by MTs at different [free tubulin] follows a sigmoidal relationship (e.g., (9, 31, 32)); (ii) detectable polymers (i.e., filaments longer than the ~200 nm diffraction limit) are rare until [free tubulin] is well above Q3 ≈ CCIndGrow (9, 31). Suggestions for the identity of the nucleation process have included conformational maturation or sheet closure (9, 31). Both of our simulations exhibit behavior similar to experimental observations (i) and (ii) (compare Figure 5A-B to Figures 5C-D and S2C-D), but neither incorporates any explicit nucleation, conformational maturation, or sheet closure processes. Therefore, it is necessary to identify additional explanations for these observations.
One possibility is that the shape of the curves in Figures 5C-D and S2C-D is simply the outcome of dynamic instability itself. When [free tubulin] is just above CCe, the growth velocity during the growth phase is low (Vg = 0 at CCe) and the frequency of catastrophe is high (Tables S1-S2). Under these conditions, individual microtubules will be both short and short-lived, and so only a small amount of polymer will accumulate. As [free tubulin] rises, MTs will experience growth phases that last longer and have faster velocity, resulting in MTs that extend further. As seen by observing the average MT length or the maximum MT length, the MT lengths rise more rapidly with increasing [free tubulin], i.e., the relationship is non-linear (Figure S2C-D). Correspondingly, the fraction of seeds occupied by MTs with lengths above the detection threshold (Figure 5C-D) would be expected to increase in a similar non-linear way and thus mimic the outcome of a nucleation step without directly involving one. Because our simplified model lacks structural detail, this explanation may be sufficient to explain the sigmoidal shape of the seed-occupancy plot in that simulation (Figure 5C).
In our detailed model, an additional process may contribute: we have observed that in this model, non-virgin seeds in the presence of [free tubulin] near CCIndGrow are frequently coated by a layer of GDP tubulin left over from the previous MT.‡ Because incoming GTP subunits will detach from these GDP subunits faster than from the GTP subunits found at the surface of a virgin seed or a growing tip, growth from a GDP-coated seed will be more difficult than extension of an already growing tip. This process could be considered to be a type of nucleation, and might be predicted to exist in physical microtubules; to our knowledge, it has not previously been considered in explanations of why growth of MTs from stable seeds appears to involve a nucleation step.
Taking all this information together, we propose that both dynamic instability itself and coating of seeds by GDP subunits contribute to the observation of nucleation-like phenomena in experimental MT growth from seeds and centrosomes (9, 31, 32). While we cannot exclude the existence of processes such as conformational maturation or sheet closure, these considerations suggest that neither hypothetical process is necessary to explain the absence of detectable MTs on seeds at [free tubulin] near CCIndGrow.
Assessing the behavior of the polymerizing MT systems via J(c) plots
To further extend and test the understanding outlined above, we compared the CC values as measured in the simulated experiments above to those of a different and less frequently used type of experiment, the so-called “J(c) plots”, where “J” (a typical abbreviation for flux) is plotted as a function of subunit concentration (see e.g., (35)) (Figure 6). The construction of a J(c) plot involves assessing the rate of change in polymer mass (also described as the flux into or out of polymer) after a population of MTs at steady state is diluted into a large pool of tubulin at a new concentration.§ In these plots, “the CC” is identified as the concentration at which the curve crosses J = 0, i.e., where net growth = 0. At this concentration, individual MTs undergo periods of growth and shortening, but the population-level fluxes into and out of polymer are balanced. One could consider a J(c) plot to be analogous to a CCe plot, but with measurements performed on a population with all behaviors included, instead of on individual filaments using only growth phases. We will call the CC as measured via J(c) plots “CCJ(c).” Strikingly, the value for CCJ(c) that we measure in our simulations (see Q4 in Figure 6) corresponds to CCPopGrow as measured by the other methods.
Thus, with the exception of Q3 (Figures 1 and 5), which provides an estimate of CCIndGrow, all other experimental measurements of CC (Figure 3, 4, 6) provide a measure of the critical concentration for persistent population growth (CCPopGrow). Only Q3 provides experimental information about the concentration of tubulin needed for individual filaments to grow transiently (CCIndGrow).
Impact of changing the number of MT seeds
Since all of the simulations above were performed with a defined number of MT nuclei (seeds), it is important to address the question of how the behaviors described above might change if the number of MT seeds is altered. Thus, we repeated the simulations of Figure 3A with different numbers of stable MT seeds. Examination of the results (Figure 7A) shows that changing the number of MT seeds does change the shape of the curves in classical CC plots. More specifically, when the number of MT seeds is small, a relatively sharp transition is seen at CCPopGrow in graphs of [free tubulin] and [MT polymer]; little if any bulk polymer is observed at [total tubulin] below CCPopGrow (Figure 7A). In contrast, when the number of MT seeds is high, measurable amounts of polymer appear at concentrations well below CCPopGrow, and [free tubulin] at steady state approaches the Q2 asymptote more gradually (Figure 7A, compare light curves to dark curves). Regardless, the value of the Q2 asymptote (and thus CCPopGrow) is not affected. These observations indicate that the number of MT seeds does not impact the value of CCPopGrow, but does affect how sharply steady-state [free tubulin] approaches CCPopGrow (see also (4)).
The relationship between CC values, the behavior of individual filaments, and the behavior of bulk polymer
How can the above conclusion that MTs grow transiently at [tubulin] between CCIndGrow and CCPopGrow be reconciled with classical experimental observations of MTs in which bulk polymer appears abruptly at Q1 (Figure 1A, e.g., (4))? As discussed above, Q1 provides a measure of CCPopGrow, but traditionally it is expected to provide the critical concentration for polymer assembly, CCPolAssem. This apparent conflict can be resolved by recognizing that the fraction of total subunits converted to polymer will be low until the free subunit concentration nears CCPopGrow because the average MT filament is very short at [free tubulin] below CCPopGrow (Figures 4, S2); under these conditions, [free tubulin] will be approximately equal to [total tubulin] unless there are many stable seeds (Figure 7A). In contrast, at [total tubulin] > CCPopGrow, all subunits in excess of CCPopGrow will be converted to polymer if sufficient time is allowed.** This will happen because the average MT filament will experience net growth until [free tubulin] falls below CCPopGrow.
The outcome of these relationships is that in bulk polymer experiments, little if any MT polymer will be detected†† until the total tubulin concentration is above CCPopGrow, even though dynamic individual MT filaments can transiently exist at the lower tubulin concentrations. Thus, the experimental values Q1 and Q2 may look like the expected critical concentration for polymer assembly CCPolAssem, even though they actually represent the critical concentration for persistent population growth (CCPopGrow).
Conclusions and Significance
Based on our analysis of the behavior of our computational simulations of microtubule dynamics, we conclude that the classical understanding of the relationship between subunit concentration and assembled polymer needs significant adjustment for dynamic instability polymers. In summary, for microtubules and other polymers that exhibit DI, there is no true CCPolAssem as classically defined. Instead, there is a lower CC (CCIndGrow), above which individual filaments can grow transiently, and an upper CC (CCPopGrow), above which a population of filaments will grow persistently, even while the individual filaments in this population may still undergo dynamic instability. What might be considered “normal” dynamic instability (where individual MTs repeatedly depolymerize back to the seeds) occurs at [free tubulin] between the two critical concentrations, CCIndGrow (estimated by Q3) and CCPopGrow (measured by Q1, Q2, Q4, Q5, and Q6). For a summary of our major conclusions, see Box 1 and Figure 7B.
Box 1: The relationship between critical concentration and dynamic instability
Instead of one “critical concentration” for polymer assembly, microtubules have at least two critical concentrations, each with unique implications for polymer behavior (table and figure below). One is the critical concentration for transient growth of individual filaments (CCIndGrow). The other is the critical concentration for persistent growth in the mass of a population of filaments (CCPopGrow).
View this table:The standard population-based experimental methods for measuring critical concentration (Q1, Q2, and Q4; see Table 1) all yield the critical concentration for persistent population growth (CCPopGrow) (Figures 3, 4, 6). Of the standard experimental measurements, only Q3 (based on analysis of individual filaments) yields an approximation (perhaps poor) of CCIndGrow (Figure 5).
What might be considered “typical” steady-state dynamic instability (where filaments eventually depolymerize back to the seed) occurs at free subunit concentrations below the value Q1 ≈ Q2 ≈ Q4 ≈ Q5 ≈ Q6, which is classically considered to be “the” CC for polymer assembly (Figure 4).
The appearance of polymers on stable seeds is smooth and non-linear with respect to [free tubulin]. Filaments may not be detectable until [free tubulin] is well above CCIndGrow because the dependence of the DI parameters on [free tubulin] dictates a non-linear relationship between MT length and [free tubulin] (Figure S2C-D), mimicking a nucleation step without necessarily involving one (Figure 5C-D).
In a closed system (where filaments compete for a limited pool of subunits), [free tubulin] that is initially above CCPopGrow will fall to a steady-state value that is below but perhaps very close to CCPopGrow (Figures 3A-B, 7A).
For such closed systems, a key conclusion is that steady-state [free tubulin] is not a constant independent of [total tubulin] as expected from traditional understanding of critical concentration, but is a variable that depends on [total tubulin] and on multiple other aspects of the system, such as the presence/absence of stable seeds (Figure 7A, see also (4)(19)).
Changes to the number of stable seeds will affect neither CCIndGrow nor CCPopGrow, but can affect both the amount of MT polymer and the separation between steady-state [free tubulin] and CCPopGrow (Figure 7A).
Relationship to previous work
In considering our conclusions, it is important to recognize that Hill and Chen previously suggested two critical concentrations for microtubule assembly (14) (15). However, the idea that dynamically unstable polymers have (at least) two critical concentrations is not widely recognized, perhaps because their work left unclear how the behaviors of individual filaments and their populations relate to each other and to the critical concentrations. Another characteristic that distinguishes our conclusions from those of Hill is that the Hill work investigated CC values that correspond most closely to our Q3 (Figure 5) and Q4 (Figure 6); the relationship between critical concentrations and the remainder of the experimental values (Figures 3, 4) that we discuss above remained undetermined. Also significant is the work of Dogterom and colleagues, who as discussed above previously predicted that there is a critical concentration at which MTs will transition between “bounded” (steady-state polymer mass) and “unbounded” (persistently growing) regimes (36, 37). However, the question of how their critical concentration for unbounded growth relates to the classical definitions of critical concentration (Table 1) was left unresolved. Our results in Figure 4C-D show that Dogterom’s critical concentration corresponds most closely to our Q5 (Q5 ≈ Q1 ≈ Q2 ≈ Q4).
Significance and biological implications
We suggest that the understanding of critical concentration as presented above will help clarify apparently contradictory results in the microtubule literature. In particular, we conclude that reported measurements of “the” critical concentration for MT polymerization vary at least in part because some experiments measure CCIndGrow (e.g., (9)), while others measure CCPopGrow. (e.g., (2)). This clarification should help in design and interpretation of experiments involving critical concentration, especially those investigating the effect of MT binding proteins (e.g. (7, 9, 38)) or drugs (e.g. (8, 39)).
More importantly, these ideas help clarify the behavior of MTs in vivo. MTs in many interphase cell types grow persistently (perhaps with catastrophe and rescue, but with net positive drift) until they reach the cell edge, where they undergo repeated cycles of catastrophe and rescue (26). We showed previously that this persistent growth is a predictable outcome of having enough tubulin in a confined space: if sufficient tubulin is present, the MTs grow long enough to contact the cell boundary, which causes catastrophe; this drives the [free tubulin] above its natural steady-state value, which reduces catastrophe, enhances rescue, and induces the persistent growth behavior (19). In light of the current results, we can now phrase this explanation more succinctly: persistent growth of MTs in interphase cells occurs when catastrophes induced by the cell boundary drive [free tubulin] above CCPopGrow. In contrast, at mitosis, when the MTs are more numerous and thus shorter, [free tubulin] remains below CCPopGrow (see also (19)(37)).
These ideas also enable a more nuanced understanding of the effects of MT binding proteins (MTBPs). MTBPs are traditionally described as being stabilizers, destabilizers, or sequestering proteins and are quantitatively characterized by their effects on DI parameters. While these descriptors remain valid, we suggest that a greater depth of understanding could be obtained by also describing a MTBP in terms of its effect on CCIndGrow and CCPopGrow. For example, a protein that “stabilizes” MTs by reducing CCIndGrow (e.g., by increasing the fraction of incoming subunits incorporated into the MT) might be expected to increase the fraction of nuclei with active MTs; such an activity would be expected to cause an accompanying decrease in average MT length in a closed system such as a cell. In contrast, a different MTBP that stabilizes MTs by decreasing CCPopGrow (e.g., by increasing rescue) should result in longer MTs that are fewer in number. The idea that two proteins both categorized as “stabilizers” could have such different effects is initially confusing, but easier to understand when viewed in the light of their different effects on CCIndGrow and CCPopGrow.
Another significant conclusion is the remarkable concurrence as seen in Figure 4C-F between three seemingly disparate ways of measuring and analyzing MT behavior: (i) the rate of change in [MT polymer] (Figure 4C-D, circles), which is a bulk property obtained by assessing the mass of the population of polymers at different points in time (e.g., across 15 minutes); (ii) the JDogterom equation (Figure 4C-D, + symbols), which uses DI parameters extracted from individual length history plots obtained over tens of minutes; (iii) the drift coefficient, which is measured from observing individual MTs in a population of MTs for short periods of time (e.g., 4 second intervals across as little as one minute see Figures 4E-F, S2E-F). These approaches differ in attributes including physical scale, temporal scale, and experimental design. The agreement between the results of these measurements suggests that the experimentally more tractable drift coefficient approach (25, 26) captures key quantitative aspects of DI and is likely to be a useful way of assessing CCPopGrow in the future.
Though the studies presented here were formulated specifically for MTs, we suggest that they can be applied to any nucleated, steady-state polymers that display dynamic instability. Moreover, we propose that the key characteristic that distinguishes dynamically unstable steady-state polymers (e.g. mammalian MTs) from other polymers (e.g., mammalian actin) is as follows: for DI polymers, CCIndGrow and CCPopGrow are separable values, but for the other polymers, they are either identical (as is true for equilibrium polymers) or so close as to be nearly superimposed (e.g., mammalian actin). The kinetic origins and implications of this difference are the topic of ongoing work.
Methods
Detailed Model
The detailed computer simulation of microtubule dynamics with parameters tuned to approximate in vitro dynamic instability of mammalian brain MTs was described previously (20, 30, 40). The core simulation is the same as that in (20, 40), but this version has minor modifications including the addition of a dilution function to enable production of J(c) plots such as those in Figure 6. Please refer to the original work (20) for detailed information on the model, its parameter set C, and how its behavior compares to that of in vitro dynamic instability.
Simplified Model
The simplified model of microtubule dynamics was also described previously (19), but the implementation used here was updated in several major ways as described in Supplementary Information. These changes mean that the observed DI parameters (Table S1) and Q values (Figures 3–7) are not directly comparable between this implementation and our earlier publication (19), but the general behavior of the simulation is the same.
Analysis
For all analysis procedures, please see Supplementary Information.
Acknowledgments
This work was supported by NSF grants MCB-0951264 and MCB-1244593 to HVG and MSA. Portions of the work were also supported by funding from the University of Massachusetts Amherst (AJM) and by a fellowship from the Dolores Zohrab Liebmann Fund (SMM). We thank the members of the Chicago Cytoskeleton community for their insightful discussions, and for members of the Goodson laboratory for assistance in editing the manuscript.
Footnotes
↵* Unless otherwise indicated, in this manuscript we use the term “steady state” to refer specifically to polymer mass steady state, which describes a situation where the polymer mass of a closed system of dynamic MTs (i.e., a system with constant [total tubulin]) has reached a plateau and no longer changes with time (other than small fluctuations around the steady-state value). Systems of dynamic microtubules can also have other steady states (e.g., polymer length steady state). Ambiguity about which steady state a system is in can cause confusion in designing and interpreting experiments.
↵† Note: here a “bounded” system refers to one that has a constant steady-state polymer mass or average length; “unbounded” refers to a system where the polymer mass or average length exhibits net growth over time (36, 37). We do not use the terms bounded and unbounded in defining the CC values to avoid possible confusion with situations where a system of MTs is or is not physically bounded by a barrier. This ambiguity can be especially confusing when discussing MT behaviors in vivo (19).
↵‡ This layer of GDP subunits becomes “stuck” on the non-hydrolyzing GTP-seed because, in our standard parameter set (20), it is the identity of the lower subunits (the ones with the exposed nucleotide) that dictate koff, consistent with MT structure (33). A GDP subunit on the seed has a low koff because it is attached to a GTP subunit, but incoming GTP subunits detach rapidly from this GDP subunit. This relationship has been termed “trans-acting GTP,” and is expected to exist in physical MTs (34).
↵§ J(c) plots are so called because they plot polymer flux (J) as a function of concentration (c). In the physical experiments, there was normally a delay of a few seconds after the dilution and before the data were recorded. This delay may have been necessary for technical reasons, but it also serves a purpose in allowing the stabilizing GTP cap to redistribute to its steady-state size. Analysis of our simulated J(c) experiments incorporates a similar delay.
↵** More precisely, as indicated by the earlier discussion of Figure 3A-B, all subunits in excess of the steady-state [free tubulin] will be converted to polymer; the steady-state [free tubulin] is necessarily below but perhaps close to CCPopGrow.
↵†† The amount of polymer present depends on the kinetic rate constants of the particular system and the number of stable seeds. The amount of polymer detected depends on the amount of polymer actually present and on what the experimental setup can detect.