## ABSTRACT

Behaviors of dynamic polymers such as microtubules and actin are frequently assessed at one or both of two scales: (i) net assembly or disassembly of bulk polymer, (ii) growth and shortening of individual filaments. Previous work has derived various forms of an equation to relate the rate of change in bulk polymer mass (i.e., flux of subunits into and out of polymer, often abbreviated as “*J*”) to individual filament behaviors. However, these versions of this “*J* equation” differ in the variables used to quantify individual filament behavior, which correspond to different experimental approaches. For example, some variants of the *J* equation use dynamic instability parameters, obtained by following particular individuals for long periods of time. Another form of the equation uses measurements from many individuals followed over short time steps. We use a combination of derivations and computer simulations that mimic experiments to (i) relate the various forms of the *J* equation to each other; (ii) determine conditions under which these *J* equation forms are and are not equivalent; and (iii) identify aspects of the measurements that can affect the accuracy of each form of the *J* equation. Improved understanding of the *J* equation and its connections to experimentally measurable quantities will contribute to efforts to build a multi-scale understanding of steady-state polymer behavior.

## 1 INTRODUCTION

Behaviors of dynamic polymers such as microtubules (MTs) and actin are frequently assessed at the scales of populations and/or individual filaments. Previous work has investigated various forms of an equation that use quantities describing individual filament dynamics to estimate the rate of change in the population’s polymer mass (e.g., (Hill & Chen, 1984; Walker et al., 1988; Verde, Dogterom, Stelzer, Karsenti, & Leibler, 1992; Dogterom & Leibler, 1993; Komarova, Vorobjev, & Borisy, 2002)). The rate of change in the population’s polymer mass is also described as the flux (abbreviated as *J*) of subunits into and out of polymer, so we refer to this equation as the *J* equation. The versions of the *J* equation differ in the particular variables used to quantify individual filament behavior, which correspond to different experimental approaches (e.g., following particular individuals for long times (Walker et al., 1988), or many individuals each over short time steps (Komarova et al., 2002)). In this paper, we relate the various forms of the *J* equation to each other and use computational simulations to demonstrate these relationships. We also discuss aspects of the measurements that can affect the accuracy of the output of each form of the *J* equation. This paper focuses on microtubules, but should apply to steady-state polymers more broadly. For definitions of abbreviations and terms used in this paper, please see Table 1.

### 1.1 Flux of subunits into and out of polymer

The flux (*J*) of subunits into and out of polymer has been used to quantify behaviors of polymers such as microtubules and actin (e.g., (Carlier, Pantaloni, & Korn, 1984b; Carlier, Hill, & Chen, 1984a; Verde et al., 1992; Vavylonis, Yang, & O’Shaughnessy, 2005). In a traditional flux measurement experiment, the relationship between flux and subunit concentration is determined by first growing filaments to long lengths in an environment with a high concentration of subunits, then “diluting” (transferring) samples into known concentrations of subunits and assessing the rate at which polymer assembles or disassembles (e.g., (Carlier et al., 1984a; Carlier et al., 1984b). Figure 1A,D shows schematic representations of *J* as a function of subunit concentration as obtained from such a dilution experiment (panel **A**) and from another type of experiment where [free subunit] is held constant for the entire duration of the experiment (panel **D**).

When the net flux of subunits into a population’s polymer mass is *positive* (*J* > 0), the average filament length *increases* over time (Figure 1B,E, squares). In contrast, when the net flux of subunits into a population’s polymer mass is *negative* (*J* < 0), the average filament length *decreases* over time (Figure 1B, circles). Polymer-mass steady state is when the net flux of subunits into a population’s polymer mass is *zero* (*J* = 0); in this situation, the average filament length stays *constant* over time (Figure 1E, diamonds). The [free subunit] above which *J* > 0 is a critical concentration (CC). This CC can be described as the [free subunit] above which “net assembly” (Walker et al., 1988) or “unbounded growth” (Dogterom & Leibler, 1993; Fygenson, Braun, & Libchaber, 1994) will occur (CC_{NetAssembly} in Figure 2) (see also (Hill & Chen, 1984; Hill, 1987; Jonasson et al., 2019).

### 1.2 Dynamic instability of individual microtubules

Dynamic instability (DI) is a behavior in which individual microtubules stochastically alternate between periods of growth and shortening (Mitchison & Kirschner, 1984) (Figure 1C,F). Transitions from growth to shortening are called catastrophes. Transitions from shortening to growth, without complete depolymerization, are called rescues. DI is commonly quantified by four parameters: growth velocity (*V*_{g}), shortening velocity (*V*_{s}), catastrophe frequency (*F*_{cat}), and rescue frequency (*F*_{res}). The [free tubulin] above which *V*_{g} > 0 is the CC above which “elongation” phases of individual filaments can occur (CC_{Elongation} in Figure 2A-B) (Hill & Chen, 1984; Hill, 1987; Walker et al., 1988; see also Jonasson et al., 2019).

### 1.3 Relationship between flux and dynamic instability

Individual MTs can display DI when *J* is positive, negative, or zero. When *J* is *positive*, individual MTs experience *net growth* (more length increase during growth than length decrease during shortening) over sufficient time (Figure 1C,F, label *J* > 0). When *J* is *negative*, individual MTs experience *net shortening* (more length decrease during shortening than length increase during growth) over sufficient time (Figure 1C, label *J* < 0). When *J* equals *zero*, individual MTs experience *no net length change* over sufficient time (Figure 1F, label *J* = 0); in this case, growth and shortening can both occur, but the length changes balance each other out.

### 1.4 Equation relating flux and dynamic instability

Previous papers have presented various forms of an equation relating dynamic instability to the flux of subunits into and out of polymer. To our knowledge, this equation was first presented in (Hill & Chen, 1984). The flux of subunits into and out of polymer for an individual microtubule over sufficient time or averaged over MTs in a sufficiently large population can be given by

(Hill & Chen, 1984; Komarova et al., 2002). *V*_{g} is the growth velocity during growth phases and *V*_{s} is the shortening velocity during shortening phases. *P*_{growth} and *P*_{shortening} are the probabilities of being in growth or shortening. *P*_{growth} can be thought of as the proportion of time in growth (Hill & Chen, 1984; Walker et al., 1988; Gliksman, Parsons, & Salmon, 1992) or the proportion of individuals that are in growth within a population (Komarova et al., 2002), and analogously for *P*_{shortening}. Note that *J* can be determined from *V*_{g}, *V*_{s}, *P*_{growth}, and *P*_{shortening} by using Equation 1, but that *V*_{g}, *V*_{s}, *P*_{growth}, and *P*_{shortening} cannot be uniquely determined from *J* alone.

We use *V*_{s} to mean the shortening *velocity* (negative number). If *V*_{s} is used to mean the shortening *speed* (positive number), then the plus sign in Equation 1 would be become a minus sign: *J* = *V*_{g} ∗ *P*_{growth} − *V*_{s} ∗ *P*_{shortening}. These sign conventions are chosen so that growth results in an increase in length and shortening results in a decrease in length.

Under conditions where *P*_{shortening} is near zero (e.g., if [free subunit] is high), *J* is approximately equal to *V*_{g} (Figure 3, dotted line). In this case, almost all individual MTs in a population would be in growth. Under conditions where *P*_{growth} is near zero (e.g., if [free subunit] is low), *J* is approximately equal to *V*_{s} (Figure 3, dashed line). In this case, almost all individual MTs in a population would be in shortening.

For a population of MTs, *J* as written in Equation 1 represents the per microtubule average flux of subunits into and out of polymer. For all equations in this paper we assume that *V*_{g}, *V*_{s}, and *J* are in units of length/time (e.g., µm/s). In this case, *J* is equivalent to the rate of change in average MT length (in this paper, our average MT length calculated as the sum of the lengths of all individuals MTs in the population divided by the number of stable MT seeds). If the right hand side of Equation 1 (or any of the subsequent *J* equations in this paper) is multiplied by the number of individuals in a population and the units are converted to concentration/time (e.g., µM/s), then *J* will represent the flux of subunits into and out of the population’s overall polymer mass, instead of the per microtubule average length. The quantity *J* in Equation 1 is also referred to as the drift coefficient, which represents the rate of displacement of the MT ends (Vorobjev, Rodionov, Maly, & Borisy, 1999; Maly, 2002; Komarova et al., 2002; Vorobjev & Maly, 2008; Mirny & Needleman, 2010).

For a system where both ends (plus and minus) of each filament are free, Equation 1 can be applied to each end separately. In this case, the rate of change in average MT length would be the sum of *J* for the plus end and *J* for the minus end. If one end of each filament is anchored (e.g., at a centrosome), then *J* for the free end is equivalent to the rate of change in average MT length. The work in this paper examines the latter case, in which MTs are active at only one end.

### 1.5 Role of flux and dynamic instability in defining critical concentrations

The relationship between *J* and *V*_{g} in the *J* equation can be useful in understanding two critical concentrations (CC_{NetAssembly} and CC_{Elongation} in Figure 2) that are relevant to the behaviors of DI polymers. CC_{NetAssembly} (called *c*_{o} in (Hill & Chen, 1984), *c*_{cr} in (Dogterom & Leibler, 1993), CC_{PopGrow} in (Jonasson et al., 2019)) is the higher of these two CCs and is the [free tubulin] above which *J* > 0. At [free tubulin] above CC_{NetAssembly}, the average MT length or polymer mass of a population will increase persistently, and individuals will experience net growth over time.

CC_{Elongation} (called *c*_{1} in (Hill & Chen, 1984), *S*_{c}^{e} in (Walker et al., 1988), CC_{IndGrow} in (Jonasson et al., 2019)) is the [free tubulin] above which *V*_{g} > 0. CC_{Elongation} is measured by extrapolation to *V*_{g} = 0 from a plot of *V*_{g} versus [free tubulin]. At [free tubulin] above CC_{Elongation}, individual MTs can exhibit transient growth phases, though for [free tubulin] near CC_{Elongation}, few MTs will exceed experimentally relevant detection thresholds (Jonasson et al., 2019).

For polymers that do not display (detectable) DI, CC_{Elongation} and CC_{NetAssembly} are either the same (e.g., equilibrium polymers) or experimentally indistinguishable (e.g., actin). For such polymers, when *J* > 0, individuals grow (*P*_{growth} ≈ 1) and *J* ≈ *V*_{g}; when J < 0, individuals shorten (*P*_{shortening} ≈ 1) and *J* ≈ *V*_{s}. In contrast, for polymers that display DI, CC_{Elongation} and CC_{NetAssembly} are distinguishable (Jonasson et al., 2019). This is case depicted in the schematics in Figures 1 to 3, and that will be examined in this paper.

### 1.6 Outline of approach

In this paper, we compare variants of the *J* equation, examine conditions under which the different forms of the equation are equivalent, and demonstrate how to convert between the forms. We will use subscripts on *J* to distinguish the specific versions of the equation. As the starting point for the analysis, we use the *J*_{General} equation (Equation 1), which depends on the probabilities of being in growth or shortening, *P*_{growth} and *P*_{shortening}, respectively (similar to (Hill & Chen, 1984; Komarova et al., 2002)). In the Results and Discussion section, we first examine forms of the equation (*J*_{Time} and *J*_{TimeStep}) in which *P*_{growth} and *P*_{shortening} are determined from the fraction of time spent in growth or shortening (e.g., (Komarova et al., 2002)). Next, versions of the equation (*J*_{DI} and *J*_{DI_piecewise}) that use *F*_{cat} and *F*_{res} to calculate *P*_{growth} and *P*_{shortening} are considered (Hill & Chen, 1984; Walker et al., 1988; Verde et al., 1992; Dogterom & Leibler, 1993).

The *J*_{DI} form of the equation is perhaps the most well-known, because Dogterom and colleagues related this form of the equation to bounded and unbounded growth behaviors (Verde et al., 1992; Dogterom & Leibler, 1993). In the “bounded” growth regime, the average MT length reaches a steady-state value over time. In the “unbounded” growth regime, the average MT length increases indefinitely. Forms of the *J* equation, most commonly *J*_{DI}, have been utilized in many other papers (e.g., (Gliksman et al., 1992; Bicout, 1997; Maly, 2002; Vorobjev & Maly, 2008; Mirny & Needleman, 2010; Yarahmadian, Barker, Zumbrun, & Shaw, 2011; Mahrooghy, Yarahmadian, Menon, Rezania, & Tuszynski, 2015; Ishihara et al., 2016; Aparna, Padinhateeri, & Das, 2017; Lamson, Edelmaier, Glaser, & Betterton, 2019; Kuo, Trottier, Mahamdeh, & Howard, 2019)).

We use our previously established computational simulations to illustrate the results of the *J* equations at various tubulin concentrations and to demonstrate how aspects of experimental design, such as the timing of the experimental steps, can lead to errors in measuring *J*.

As discussed above, the variants of the *J* equation use measurements on individuals to calculate estimates of *J*. To assess these variants, we compared each to *J*_{Net}, which is the *net* rate of change in average MT length as calculated directly from the change in the average MT length of the population between two time points. *J*_{Net} provides the true net rate of change that occurs in any particular run of the simulation, and is therefore a useful baseline for comparison.

## 2 RESULTS AND DISCUSSION

### 2.1 Computational Simulations

In our dimer-scale computational model (introduced in (Margolin, Goodson, & Alber, 2011; Margolin et al., 2012)), the attachment and detachment of subunits from protofilaments, the formation and breakage of lateral bonds between protofilaments, and the hydrolysis of subunits from GTP-tubulin to GDP-tubulin are modeled as discrete events. The biochemical kinetic rate constants for these processes are inputted by the user. The values of the rate constants used here were tuned in (Margolin et al., 2012) to approximate the plus-end dynamics of mammalian MTs as reported in (Walker et al., 1988). The MTs grow from stable non-hydrolyzable GTP-tubulin seeds, and all attachment and detachment events occur at the free end of each MT. The values of *J* and the DI parameters are emergent properties of the system. This is analogous to experimental systems, where the values of *J* and the DI parameters will depend on cell type and experimental conditions such as the source of the tubulin and the buffer. For additional details about the simulations, please see the Methods, Section 3.1.

We use our computational model to simulate dilution experiments and constant [free tubulin] experiments (see Figure 1 for schematic representations of *J* and DI behaviors in these types of experiments). Except where otherwise indicated, all simulations in this paper were performed with a population of 50 stable MT seeds.

### 2.2 Time-based *J* Equation

One way to calculate *P*_{growth} and *P*_{shortening} to use in Equation 1 is
and

Then *J* can be calculated as

One method for obtaining values of *V*_{g}, *V*_{s}, total time in growth, and total time in shortening is to use standard DI analysis to identify periods of growth and shortening in length histories (see Methods, Section 3.2.2). Figure 4 shows plots of *V*_{g}, *V*_{s} (panels **A,C**), *P*_{growth}, and *P*_{shortening} (panels **B,D**) as obtained from the DI analysis, as well as *J*_{Net} for comparison (panels **A,C**). At very high [free tubulin] in both the dilution and constant [free tubulin] simulations, *P*_{shortening} ≈ 0, *P*_{growth} ≈ 1, and J ≈ *V*_{g}. In contrast, in the dilution simulations, at very low post-dilution [free tubulin], *P*_{shortening} ≈ 1, *P*_{growth} ≈ 0, and *J* ≈ *V*_{s}. However, in the constant [free tubulin] simulations, at very low [free tubulin], *P*_{shortening} ≈ *P*_{growth} ≈ 0, and J ≈ 0.

The difference at low [free tubulin] between the dilution and constant [free tubulin] systems occurs because the MTs in dilution systems are sufficiently long as to not undergo complete depolymerizations back to the seed during the measurement period (Figure 5A-B); in contrast, the MTs in constant [free tubulin] systems are short at low tubulin and therefore frequently and repeatedly depolymerize back to the seed (Figure 5C-D).

Figure 6 shows the results of the *J*_{Time} equation (Equation 4) evaluated with *V*_{g}, *V*_{s}, total time in growth, and total time in shortening as measured with our DI analysis method (Methods, Section 3.2.2). The *J*_{Time} results match well with the direct measurements of *J* from the net rate of change in average MT length, *J*_{Net}, in the simulations data (Figure 6).

When applying the *J*_{Time} equation (Equation 4), one should be aware of the possibility of other states besides growth and shortening. In Equations 2, 3, and 4, “total time in all phases” is the time of all observations and can include time in other phases (e.g., pause) during which the MT length is approximately constant. Thus, total time may be greater than (total time in growth) + (total time in shortening). If (total time in growth) + (total time in shortening) were used in place of total time in Equation 4 and if the MTs spent some time in an additional state such as pause, then the equation would overestimate the magnitude of the actual rate of change in average length. However, the sign of *J* would not be affected, so the equation could still be used to determine if the average length was increasing (*J* > 0), decreasing (*J* < 0), or constant (*J* = 0).

### 2.3 Time-step method for measuring *J*

An alternative approach to determine the value of *J* (called the drift coefficient and abbreviated as *v*_{d} in (Komarova et al., 2002)) is to measure the displacements of the MT ends over short time steps (e.g., between successive images in a time series); then
where **∑***s*_{i} is the sum of all the displacements of MT ends and **∑***t*_{i} is the sum of the corresponding time changes (Komarova et al., 2002). Figure 7 compares the results of the *J*_{TimeStep} equation and the *J*_{Net} data (*J*_{Net}, as above, is measured directly from the net rate of change in average MT length); the results of the two methods agree well. For implementation details of the time-step analysis method, please see the Methods, Section 3.2.3.

#### 2.3.1 Time steps during which a displacement is zero (*s*_{i} = 0)

A factor to be aware of when measuring *J* by the *J*_{TimeStep} approach is whether the experimental method can track displacements of zero (*s*_{i} = 0). To analyze this situation, let **∑**_{pos}, **∑**_{neg}, and **∑**_{zero}, respectively, represent sums (of displacements, or of the corresponding times) when the displacements are positive (*s*_{i} > 0), negative (*s*_{i} < 0), or zero (*s*_{i} = 0). For example, **∑**_{zero} *t*_{i} represents the sums of time steps during which the displacement is zero (*s*_{i} = 0). Then **∑**_{zero} *s*_{i} = 0, so **∑***Si* = **∑**_{pos} *s*_{i} + **∑**_{neg} *s*_{i}; in other words, any displacements equaling zero (*s*_{i} = 0) do not affect the value of **∑***s*_{i}. However, **∑***t*_{i} = **∑**_{pos} *t*_{i} + **∑**_{neg} *t*_{i} + **∑**_{zero} *t*_{i}, so **∑**_{zero} *t*_{i} can affect **∑***t*_{i}. If the experimental method used does not detect displacements of zero, then **∑***t*_{i} may be underestimated and therefore the magnitude of *J* would be overestimated. The relevance to any specific system would depend on whether there are displacements of zero and how often they occur.

#### 2.3.2 Mathematical equivalence of *J*_{TimeStep} (Equation 5) to *J*_{General} (Equation 1) and *J*_{Time} (Equation 4)

To see the equivalence of *J*_{Time} (Equation 4) and *J*_{TimeStep} (Equation 5), the displacements can be separated into positive and negative displacements. *V*_{g} and time in growth are determined from the positive displacements, while *V*_{s} and time in shortening are determined from the negative displacements. More specifically, *V*_{g} is **∑**_{pos} *s*_{i} / **∑**_{pos} *t*_{i} where the sums include only the positive displacements;**∑**_{pos} *t*_{i} is the time in growth; then

Similarly, *V*_{s} = **∑**_{neg} *s*_{i} / **∑**_{neg} *t*_{i} where the sums include only the negtive displacements; **∑**_{neg} *t*_{i} = time in shortening; and

Also, **∑** *s*_{i} = **∑**_{pos} *s*_{i} + **∑**_{neg} *s*_{i}. Then,

Thus, *J*_{TimeStep} (Equation 5) is algebraically equivalent to *J*_{Time} (Equation 4), and therefore to *J*_{General} (Equation 1).

Alternatively, the equivalence of *J*_{General} (Equation 1) and *J*_{TimeStep} (Equation 5) can be shown using *P*_{growth} = **∑**_{pos} *t*_{i} / **∑** *t*_{i} and *P*_{shortening} = **∑**_{neg} *t*_{i} / **∑***t*_{i}, yielding

Note that *V*_{g} and *V*_{s} as calculated from the time-step method depend on the size of the time-step (see Supplement Methods of (Jonasson et al., 2019)), and may differ from *V*_{g} and *V*_{s} as calculated from the DI analysis method described above (Section 2.2). However, the results of the *J*_{Time} or *J*_{TimeStep} equations will still fit well with the data, as long as the velocities and the probabilities are determined in a way that is internally consistent (Figure 8).

### 2.4 Calculating *J* from the DI parameters

As indicated above, Equations 1 through 5 can work even if there is time in phases during which the MT length does not change (e.g., if a pause occurs or if a MT seed is empty for some amount of time). The remaining equations in this paper will depend on the three simplifying assumptions listed below, in order to obtain forms of the *J* equation that are common in the literature (e.g., (Hill & Chen, 1984; Walker et al., 1988; Dogterom & Leibler, 1993)). However, as will be discussed below, there is later experimental evidence indicating that physical MTs may deviate from some of these assumptions (Tran, Walker, & Salmon, 1997; Jánosi, Chrétien, & Flyvbjerg, 2002; Odde, Cassimeris, & Buettner, 1995; Gardner, Zanic, Gell, Bormuth, & Howard, 2011), which could cause complications when applying the equations.

Let

In our analysis, we calculate *F*_{cat} as (number of catastrophes)/(total time in growth) and *F*_{res} as (number of rescues)/(total time in shortening).

*Assumption* (i): Individual microtubule assembly/disassembly behavior is purely a two-state process where the two states are growth and shortening.

*Assumption* (ii): *t*_{g} = 1 /*F*_{cat}.

*Assumption* (iii): *t*_{s} = 1 /*F*_{res}.

#### 2.4.1 Derivation of *J*_{AverageDuration} using Assumption (i)

Under the two-state assumption (Assumption (i)), *P*_{growth} = *t*_{g}/(*t*_{g} + *t*_{s}) and *P*_{shortening} = *t*_{s}/(*t*_{g} + *t*_{s}). Substituting these formulas for *P*_{growth} and *P*_{shortening} into Equation 1 leads to
(Walker et al., 1988; Gliksman et al., 1992; Vorobjev & Maly, 2008).

#### 2.4.2 Derivation of *J*_{DI} using Assumpt ions (i), (ii), (iii)

*P*_{growth} and *P*_{shortening} can be calculated from the frequencies of catastrophe and rescue, if Assumptions (ii) and (iii) (Hill & Chen, 1984; Walker et al., 1988) are satisfied in addition to Assumption (i) above.

Under the Assumptions (i), (ii), and (iii), and

Substituting Equations 7 and 8 into Equation 1 yields

(Hill & Chen, 1984; Walker et al., 1988; Verde et al., 1992; Dogterom & Leibler, 1993; Maly, 2002). Maly also presents a drift coefficient equation (i.e., a *J* equation) that incorporates pauses, in addition to growth and shortening (Maly, 2002).

For the dilution simulations, Figure 9 compares the results of the *J*_{DI} equation and *J* measured directly from the net rate of change in average MT length, *J*_{Net}, plotted as functions of post-dilution [free tubulin] (see also **Supplemental Figure S1**). The simulation results show that the *J*_{DI} equation fits the *J*_{Net} data fairly well, but that some deviation occurs in the [free tubulin] range from approximately 8 to 11 µM. This deviation in the intermediate range of [free tubulin] decreases with time after the dilution (Figure 9, compare **A** to **C** and **B** to **D**), but performing the measurements at such late times in physical experiments may be affected by complete depolymerizations (as occurred in Figure 9C at low [free tubulin]; further examined in Section 2.6.2).

#### 2.4.3 Complications in applying the *J*_{DI} equation to experimental systems

Recall that the derivation of the *J*_{DI} equation (Equation 9) depends on the simplifying Assumptions (i), (ii), and (iii) listed above. However, there is experimental evidence that physical microtubule behavior may deviate from these assumptions. This could cause complications in measuring the DI parameters to input into the equation.

Assumption (i) presumes that MTs do not exhibit any additional states besides growth and shortening. However, for example, (Tran et al., 1997; Jánosi et al., 2002) indicate the existence of a third state that is intermediate between growth and shortening. Detection of intermediate states in our simulations will be investigated in future work. For the analysis in this paper, we divided the length histories of individual MTs into only growth, shortening, or empty seed phases (the empty seed state is relevant when complete depolymerizations occur, as will be examined in Section 2.5).

The simplest scenario in which Assumptions (ii) and (iii) would hold is if the transition from growth to shortening is a first-order process with transition rate constant *F*_{cat}, and the transition from shortening to growth is a first-order process with transition rate constant *F*_{res}. Here, first-order means that the times until catastrophe for growing MTs and the times until rescue for shortening MTs are each exponentially distributed. In this case, the overall rate of catastrophe for the population is (*F*_{cat})*(# of growing MTs) and overall rate of rescue is (*F*_{res})*(# of shortening MTs).

However, there is evidence that times until catastrophe are not exponentially distributed, but instead follow a gamma distribution due to age-dependent catastrophe (e.g., (Odde et al., 1995; Gardner et al., 2011; Coombes, Yamamoto, Kenzie, Odde, & Gardner, 2013)). In this case, *F*_{cat} would be time-dependent; specifically the value of *F*_{cat} would increase over time during a growth phase (Gardner et al., 2011).

In our analysis, we calculated *F*_{cat} as (number of catastrophes)/(total time in growth). This provides an average *F*_{cat} value for the time period during which measurements were taken. If *F*_{cat} is age-dependent and the measurements were only taken early during growth phases, then the average *F*_{cat} would be underestimated.

### 2.5 Effect of complete depolymerizations

One situation in which Assumption (iii) fails is if MTs completely depolymerize. In this case, the transition from shortening to growth can occur by way of complete deploymerization followed by regrowth from the stable MT seed, rather than occurring only through rescue. Then, the transition frequency from shortening to growth would not be simply *F*_{res}, and Equations 7 to 9 would not hold. If there is time between the end of complete deploymerization and the start of re-growth, then Assumption (i) and Equation 6 also break down.

One case where complete depolymerizations occur is if [free tubulin] is in the range where *J* = 0 in the constant [free tubulin] simulations (Figures 4C-D and 5C-D) (Jonasson et al., 2019). The [free tubulin] at which the transition from *J* = 0 to *J* > 0 occurs in the constant [free tubulin] simulations (Figure 4C, diamonds) is CC_{NetAssembly} (Figure 2, Table 1). CC_{NetAssembly} is also the [free tubulin] at which the transition from *J* < 0 to *J* > 0 occurs in the dilution simulation (Figure 4A, circles).

As would be expected, Figure 10 shows that the *J*_{DI} equation does not fit the *J*_{Net} data from the constant [free tubulin] simulations in the [free tubulin] range where *J* = 0 (i.e., [free tubulin] below CC_{NetAssembly}).

For constant [free tubulin] < CC_{NetAssembly}, the average MT length will reach a steady-state value over time and the rate of change in average MT length will then be zero (*J* = 0) (Figures 4C and 5C). Thus, for constant [free tubulin] experiments or simulations, the *J*_{DI} equation (Equation 9) does not hold below CC_{NetAssembly} (Figure 10). Instead,
(Dogterom & Leibler, 1993). In the terminology of Dogterom et al., MTs exhibit “bounded growth” when *J* = 0 and “unbounded growth” when *J* > 0 (Dogterom & Leibler, 1993).

Depending on the specific application, empty and non-empty seeds may be considered separately; for example, in a system at polymer-mass steady state, *J* would be zero for the overall population of seeds, but would be positive for the empty seeds (since they cannot have shortening) and negative for the non-empty seeds (Vorobjev & Maly, 2008). In this case, Equations 6 to 9 could be applied to the population of non-empty seeds.

### 2.6 Timing of experimental steps and measurements in dilution systems

The accuracy of the measurement of *J* in dilution experiments can be affected by the timing of experimental steps.

#### 2.6.1 Delay for GTP cap to adjust to post-dilution [free tubulin]

After the time of the dilution, a delay before the start of the measurement period allows the GTP cap to adjust to the post-dilution [free tubulin]. Without the delay, *J* would be misestimated relative to its steady-state value, particularly at low values of [free tubulin] (Figure 11, **Supplemental Figure S2**).

#### 2.6.2 Effect of complete depolymerizations on *J*

Ideally, dilution experiments should be performed so that measurements of *J* after the dilution can be taken before any MTs have completely depolymerized to the seed. If MTs that are too short are present at the time of the dilution, they will completely depolymerize during the measurement period, causing the lower arm of *J* to shift upwards. The lengths of the MTs during the measurement period can be affected by various factors of the experimental setup (e.g., pre-dilution [total tubulin], number of seeds) and timing of the experimental steps (e.g., time of dilution, start time of the measurement period, time duration of the measurement period). To test the effects of these factors on *J*, we performed simulations with different values of pre-dilution [total tubulin] and different times of dilution. For each, we measured *J*_{Net} over varying times periods. The results in Figures 12, 13, and 14 show that there are several different factors that each increase the number of complete depolymerizations that occur during the measurement period and that all can cause the lower arm of J to shift upwards:

later of time of dilution (compare across the three columns in Figure 12);

later start time of the measurement period (compare the two data series within each of panels

**D,E,F**in Figure 12);lower pre-dilution [total tubulin] (compare panels

**A,C**to**B,D**within each of Figures 13 and 14);longer duration of the measurement period (compare the three data series within each of panels

**C,D**in Figure 14).

Traditionally, dilution experiments are performed by growing MTs to polymer-mass steady state under competing conditions (constant [tubulin total]) and then transferring the MTs to new concentrations of [free tubulin]. However, as shown in Figure 15, some MTs in a population will begin to have noticeable shortening phases even before the pre-dilution competing system has reached polymer-mass steady state. Furthermore, the longer the system is allowed to run after reaching polymer-mass steady state, the more MTs will have undergone complete depolymerization (Figure 12). In contrast, if the dilution is performed before the competing system has reach polymer-mass steady state, this can increase the time period after the dilution before any MTs completely depolymerize (Figure 13).

Thus, although one might think that it would be better to allow the system to run for a longer duration of time before the dilution, this is true only to a point. After some amount of time, MT lengths start to redistribute toward an exponential-like length distribution (e.g., Figure 15E-F; see also (Fygenson, Braun, & Libchaber, 1994)). If the dilution is performed later time, more MTs will be short and the lower arm of the measured *J* curve will increasingly be shifted upwards because of the impact of complete depolymerizations (Figure 12, compare progression from first column to last column).

### 2.7 Timing of measurements in constant [free tubulin] systems

As discussed above, measurements of *J* in dilution experiments are sensitive to the timing of experimental steps. Similarly, in constant [free tubulin] systems, *J* also depends on when the measurements are performed. Specifically, if measurements of *J* are performed too early in time, then *J* will be overestimated, particularly for [free tubulin] near CC_{NetAssembly} (the [free tubulin] at which the steady-state *J* transitions from being zero to being positive), as discussed in (Jonasson et al., 2019) and illustrated here in Figure 16. To obtain the correct steady-state value of *J*, the measurements should be performed after the system has reached polymer-mass steady state (for [free tubulin] < CC_{NetAssembly}) or polymer-growth steady state (for [free tubulin] > CC_{NetAssembly}). When using measurements of *J* to determine the value of CC_{NetAssembly}, the measurements must be taken when *J* has reached its steady-state value; reaching this state will take longer the closer [free tubulin] is to CC_{NetAssembly}.

If measurements are performed when *J* has reached its steady-state value in dilution experiments and in constant [free tubulin] experiments, then *J* from the two types of experiments will be superimposed for [free tubulin] > CC_{NetAssembly} (see Figure 6C-D of (Jonasson et al., 2019)).

### 2.8 Summary and Practical Implications

#### 2.8.1 Various forms of the *J* equation relate individual and population dynamics

The *J* equation relates the flux of subunits into and out of polymer (or rate of change in average filament length) to growth and shortening behaviors of individual MTs. To our knowledge, versions of this equation were first presented by (Hill & Chen, 1984). Since then, varied forms of the *J* equation have appeared in the literature; these forms differ in attributes including the types of experimental data used as input. From looking at variants of the *J* equation (e.g., *J*_{General}, *J*_{Time}, *J*_{TimeStep}, *J*_{DI}), it might not be obvious how they relate to each other (i.e., how to convert between different forms). Additionally, even for the same version of the equation, different authors have used many different variable names for the quantities in the equation. We show how to algebraically convert between different forms of the equation and examine the assumptions needed for the forms to be equivalent. Specifically, the *J*_{Time} and *J*_{TimeStep} equations are algebraically equivalent to the *J*_{DI} equation if the following assumptions are met: (i) individual microtubule assembly/disassembly behavior is purely a two-state process where the two states are growth and shortening; (ii) the average duration of a growth phase equals 1/*F*_{cat}; and (iii) the average duration of a shortening phase equals 1/*F*_{res}.

By using the *J* equation, measurements on individual MTs (inputs into the equation, which vary among the different forms) can be used to calculate the population-level flux behavior (output of the equation). Since it is technically difficult to measure individual-level and population-level behaviors simultaneously in physical experiments, use of the *J* equation enables one to obtain information about both scales from measurements only at the individual scale. However, correct application of any form of the equation requires understanding the conditions under which that form holds and understanding how experimental design and execution can affect the measurements, which we illustrate with computational simulations of experiments.

#### 2.8.2 Comparing versions of the *J* Equation for their validity and usefulness

The variants of the *J* equation differ in the specific measurements on individuals that are used to evaluate the equation.

For example, some of the variants use dynamic instability parameters, obtained by following particular individuals for long periods of time. In particular, the *J*_{Time} equation (Equation 4) uses *V*_{g}, *V*_{s}, total time in growth, and total time in shortening. The *J*_{DI} equation (Equation 9) also uses *V*_{g} and V_{s}, but calculates the probabilities of growth and shortening using *F*_{cat} and *F*_{res} instead of from the total times in growth and shortening (Hill & Chen, 1984; Walker et al., 1988; Verde et al., 1992; Dogterom & Leibler, 1993; Maly, 2002).

In contrast, the *J*_{TimeStep} equation (Equation 5), which is the drift coefficient formula of (Komarova et al., 2002), uses displacements of many individuals followed over short time steps and does not require the same individuals to be followed over long periods of time. Thus, experimentalists can choose the type of measurement that is most feasible for their experiments and then use the corresponding form of the *J* equation.

To assess the utility and accuracy of the above variants of the *J* equation under different conditions, we compared each to the value of *J*_{Net} as observed in our simulations. *J*_{Net} is the net rate of change in average MT length between two time points. For simulation data, *J*_{Net} provides the true net rate of change that occurs in any particular run of the simulation, and is therefore a useful baseline for comparisons.

As observed in the dilution simulations, the *J*_{Time} and *J*_{TimeStep} equations are less sensitive to the measurement time period than is the *J*_{DI} equation. Specifically, the results of both the *J*_{Time} and *J*_{TimeStep} equations closely match the *J*_{Net} data (Figures 6A and 7A). In contrast, the results of the *J*_{DI} equation deviate from the *J*_{Net} data if the measurements are taken too soon after the dilution (Figure 9, compare panels **A-B** to **C-D**). Additionally, if the *J*_{DI} measurement period is too short, few transitions will be detected, and the output of the *J*_{DI} equation will be very noisy.

As observed in the constant [free tubulin] simulations, the *J*_{Time} and *J*_{TimeStep} equations match the *J*_{Net} data even if there are complete depolymerizations (Figures 6B and 7B), but the *J*_{DI} equation does not (Figure 10). The *J*_{DI} equation uses the rescue frequency, *F*_{res}, as the rate of transitioning from shortening to growth. When there are complete depolymerizations, transitions from shortening to growth can occur not only by rescue but also by re-growth from the MT seed following a complete depolymerization. The *J*_{DI} equation does not hold in this case, because the overall rate of transition from shortening to growth is greater than *F*_{res}. Instead J_{DI_piecewise} holds (Equation 10; see also (Dogterom & Leibler, 1993)).

In both of the above cases, *J*_{Time} fits *J*_{Net} (Figure 6) better than *J*_{DI} fits *J*_{Net} (Figures 9, 10). Interestingly, the *J*_{Time} and *J*_{DI} equations both use measurements from DI analysis. The *J*_{DI} equation uses *F*_{cat} and *F*_{res}, which are calculated from the numbers of catastrophes and rescues divided by the total times in growth and shortening, respectively. If the measurement period is not long enough to capture a significant number of transitions, then *F*_{cat} and *F*_{res} can be inaccurate and imprecise. The *J*_{Time} equation also uses total times in growth and shortening, but does not require knowing the number of transitions. Since *J*_{Time} fits the data better than *J*_{DI}, *J*_{Time} may provide a more experimentally accurate way to determine *J*. More specifically, if one is performing DI analysis, then using the total time in growth and shortening directly in *J*_{Time} may provide a more accurate estimate of *J* than using total time in growth and shortening to calculate *F*_{cat} and *F*_{res} and then using *F*_{cat} and *F*_{res} in *J*_{DI}.

As mentioned earlier, the DI analysis used in both in *J*_{Time} and *J*_{DI} requires following specific individuals over long times, whereas *J*_{TimeStep} uses the displacements of many individuals over short time steps and does not require following the same individuals for long periods of time. Moreover, *J*_{TimeStep} fits *J*_{Net} (Figure 7) as closely as *J*_{Time} fits *J*_{Net} (Figure 6). Thus, *J*_{TimeStep} may be more practical than *J*_{Time} or *J*_{DI}.

*J*_{Time} and *J*_{TimeStep} work without needing to measure *F*_{cat} and *F*_{res}. However, if one can obtain accurate measurements of *F*_{cat} and *F*_{res}, then they provide information about the dynamicity of individual MTs within a population that is not provided by *J* itself (e.g., (Hill & Chen, 1984; Verde et al., 1992; Dogterom & Leibler, 1993)). Another quantity that also provides information about dynamicity is the “diffusion coefficient” of the MT lengths (e.g., (Hill, 1987; Verde et al., 1992; Dogterom & Leibler, 1993; Komarova et al., 2002; Vavylonis, Yang, & O’Shaughnessy, 2005; Mirny & Needleman, 2010)).

#### 2.8.3 Considerations specific to implementing dilution experiments

For dilution experiments, the accuracy of the measurement of *J* can be affected by various factors such as experimental set up (e.g., tubulin concentration, number of seeds) and timing of experimental steps (e.g., dilution time, measurement time period).

The MT lengths at the time of dilution depend on the pre-dilution [total tubulin], the number of seeds, and the point in time when the dilution is performed (Figures 12A-C, 14A-B, 15). If any MTs are too short at the time of dilution, complete depolymerizations occur soon after the dilution (Figure 13) and cause the lower arm of *J* to shift upwards (Figures 12D-F, 14C-D). To avoid short MTs, one might have expected that the dilution should be performed after the pre-dilution competing system has reached polymer-mass steady state and that waiting longer before the dilution would lead to longer MTs. However, as shown by Figures 12, 13, and 15 together, the ideal time to perform the dilution is before polymer-mass steady state because the proportion of short MTs in a population increases after this ideal time.

Additionally, to allow the GTP cap to adjust to the post-dilution [free tubulin] (Duellberg, Cade, Holmes, & Surrey, 2016), a delay is needed before beginning measurements after the dilution (Figures 11, **S2**). However, if the end of the measurement period is too late in time after dilution, complete depolymerizations will occur (Figures 12, 13, 14).

Thus, in designing a dilution experiment, it is necessary to account for both the requirement for a delay and the need to avoid complete depolymerizations during the measurement period. An experimental design that maximizes the length of the shortest microtubules in a population at the time of dilution will be most likely to lead to accurate measurements of *J*.

#### 2.8.4 Broader implications of the *J* equation for steady-state polymers

The *J* equation provides a way to understand how dynamic instability relates to the critical concentrations CC_{Elongation} and CC_{NetAssembly} (Figure 2). CC_{Elongation} is the [free subunit] above which *V*_{g} is positive, whereas CC_{NetAssembly} is the [free subunit] above which the steady-state value of *J* is positive. We have previously proposed that the separation between CC_{Elongation} and CC_{NetAssembly} may account for the behavioral differences between MTs and actin (Jonasson et al., 2019).

For polymer types that display instability, such as MTs, there are values of [free subunit] at which growth and shortening occur simultaneously within a population (i.e., *P*_{shortening} and *P*_{growth} are both positive). Then, as seen from *J*_{general} = *V*_{g} ∗ *P*_{growth} + *V*_{s} ∗ *P*_{shortening} (Equation 1), *V*_{g} and *J* will be different from each other (Figures 3,4). CC_{Elongation} and CC_{NetAssembly} will therefore be different from each other (Figure 2).

In contrast, for polymer types, such as actin, that do not display (detectable) dynamic instability, either *P*_{shortening} ≈ 1 or *P*_{growth} ≈ 1 at any particular subunit concentration. In other words, the population and individuals will have the same behavior, i.e., all individuals are growing or all individuals are shortening. In this case, *J* ≈ *V*_{g} whenever *J* > 0, and CC_{Elongation} and CC_{NetAssembly} would be the same or experimentally indistinguishable.

## 3 METHODS

### 3.1 Simulations

The simulations in this paper used the dimer-scale computational model of MT dynamics that was originally introduced in (Margolin et al., 2011; Margolin et al., 2012). Specifically, expect for minor changes in the amount of information being outputted, we used the same implementation of the simulation that was used in (Jonasson et al., 2019), therein referred to as the “detailed model”.

In this dimer-scale model, each MT is composed of 13 protofilaments, and each protofilament is a chain of discrete subunits representing tubulin dimers. The MT length is the average of the 13 protofilament lengths, with 1 subunit length equaling 8 nm. In the simulations, there is no physical boundary that would limit MT lengths.

The biochemical events in the model are attachment and detachment of subunits to/from protofilament tips, formation and breakage of lateral bonds between adjacent subunits in neighboring protofilaments, and hydrolysis of GTP-tubulin subunits to GDP-tubulin subunits. The kinetic rate constants for these processes are user-inputted values and depend on the nucleotide state of the subunits involved in each event. All attachment and detachment event occur at the tips of the protofilaments. One subunit can attach to a tip at a time, and any subunit or oligomer of subunits that is not laterally bonded to a neighboring protofilament can detach from a tip.

The length change behavior of an individual MT over time is an emergent property, resulting from the execution of the kinetic events described above. Consequently, the values of the DI parameters, [polymerized tubulin], and *J* are also emergent properties. Additionally, in competing systems, the value of [free tubulin] is another emergent property.

In this paper, all simulations were performed in a volume of 500 fL (= 5.00 × 10^{−13} L) with MTs growing from 50 stable non-hydrolyzable GTP-tubulin seeds, except in Figure 15C,F, which has 200 seeds. We used the kinetic rate constants from Parameter Set C of (Margolin et al., 2012), which was tuned to approximately match the plus-end dynamics of mammalian MTs at 10 µM as reported in (Walker et al., 1988). This parameter set was also used in (Jonasson et al., 2019).

Please see Table 1 for descriptions of the types of simulations: competing; non-competing or constant [free tubulin]; and dilution.

### 3.2 Analysis

#### 3.2.1 Calculation of *J*_{Net}

The net rate of change in average MT length is determined from

In the simulation outputs, the average MT length is the average of the individual MT lengths for all MT seeds in the population, and the length of each individual MT is the average of its 13 protofilament lengths.

Thus, the rate of change in average MT length can be converted to the rate of change in [polymerized tubulin] as follows:

#### 3.2.2 DI analysis method

To identify growth and shortening phases in the MT length histories and to calculate the DI parameters, we use an automated DI analysis method (presented in the Supplemental Methods of (Jonasson et al., 2019)). Briefly, the DI analysis method identifies peaks and valleys in the data such that the length change between a peak and neighboring valley is greater than or equal to a user-defined threshold. For the analysis in this paper, we set the threshold to 25 subunit lengths (200 nm) to be comparable to detection limits in typical light microscopy. The ascent from a valley to a peak is classified as a growth phase and the descent from a peak to a valley is classified as a shortening phases. The DI parameters are calculated by
*V*_{g} = total length change during growth phases / total time in growth phases,*V*_{s} = total length change during shortening phases / total time in shortening phases,*F*_{cat} = total number of catastrophes / total time in growth phases,*F*_{res} = total number of rescues / total time in shortening phases.

For more detailed information about the DI analysis method, please see the Supplemental Methods of (Jonasson et al., 2019).

#### 3.2.3 Time-step analysis method

The drift coefficient (*v*_{d}) formula of (Komarova et al., 2002) provides the basis for the time-step analysis method used to evaluate the *J*_{TimeStep} = 𝑣_{d} = (∑*s*_{i})/(∑*t*_{i}) (Equation 5). Here, we implemented the analysis as described in the Supplemental Methods of (Jonasson et al., 2019). Briefly, to apply the *J*_{TimeStep} equation to our simulation data, the length history of each MT was divided into 1-second time steps (*t*_{i}), and the displacement (*s*_{i}) of the MT ends over each time step was recorded. The displacements and corresponding time steps were then summed over all individuals and over the total measurement period. For the simulation data, the lengths of all individuals are known at all times, so ∑ *t*_{i} = (number of MT seeds) * (total time of measurement). To apply the *J*_{TimeStep} equation to experimental data, it is not necessary for the same individuals to be observable over all time steps; for such data, the sums would include only those displacements that are observed. For additional information about our time-step analysis, please see the Supplemental Methods of (Jonasson et al., 2019).

## Data Availability Statement

The simulation code, analysis codes, and data are available from the corresponding author upon request.

## Acknowledgments

This work was supported by NSF grants MCB-1244593 and MCB-1817966 to HVG and MCB-1817632 to EMJ.