The theoretical description of the polymerisation of proteins such as actin and tubulin to yield functional biostructures was considered in the 1960s [8]. For a system that evolves through primary nucleation of new filaments, elongation of existing filaments, and depolymerisation from the filament ends, the change in concentration of filaments of size j, denoted f(j; t), is given by the master equation [1,8]:

where k₊, k_off, k_n are rate constants describing the elongation, depolymerisation and nucleation steps and m(t) is the concentration of free monomeric protein in solution. The factor of 2 in Equation (1) originates from the assumption of growth from both ends. The first two terms in Equation (1) describe the change in the population of filaments of size j due to monomer addition, whilst the third and fourth terms relate to monomer dissociation. The final term in Equation (1) describes the creation of a growth nucleus consisting of n_c monomers at a rate proportional to the monomer concentration raised to this power. In addition to the direct coalescence of n_c monomers, pre-equilibrium and other schemes also result in the homogeneous nucleation term taking this form [14–25].

For the case of irreversible biofilament growth, the polymerisation rate dominates over the depolymerisation rate; from Equation (1), the rate of change of the number of filaments, P(t), and the free monomer concentration, m(t), were shown [1,8] under these conditions to obey:

Combining Equations (2) and (3) yields a differential equation for the free monomer concentration [1]:

Here, we integrate these equations in the general case where the initial state of the system can consist of any proportion of monomeric and fibrillar material; this calculation generalises the results presented by Oosawa to include a finite concentration of seed material present at the start of the reaction. Beginning with Equations (2) and (3), the substitution z(t) := log(m(t)) followed by multiplication through by dz/dt yields:

Integrating both sides results in:

This is a separable equation for dz/dt, for which the solution is given by:

Integration and exponentiation yields the expression for m(t):

The values of the constants A and B are fixed by inserting the appropriate boundary conditions in terms of m(0) and P(0). The mass concentration of polymers, M(t), is then provided through conservation of mass, M(t) = m_tot − m(t) with m_tot being the total monomer concentration. This results in the exact integrated rate law:

where the effective rate constant λ is given by λ = 2 k n k + m ( 0 ) n c and β = 2/n_c, μ = 1 + γ 2, ν = arsinh (γ) for γ = 2 k + P ( 0 ) / ( β 1 2 λ ).

We note that this expression only depends on two combinations of the microscopic rate constants, k₀ = 2k₊P(0) and λ. The result reveals that λ controls the aggregation resulting from the newly formed aggregates, whereas k₀ defines growth from the pre-formed seed structures initially present in solution. In the special case of the aggregation reaction starting with purely soluble proteins, P(0) = 0, m(0) = m_tot, these expressions reduce to μ → 1 and ν → 0, and Equation (9) yields the result presented by Oosawa [1] and the single relevant parameter in the rate equations is λ. Interestingly, generalisations of Equation (9) which include secondary pathways, maintain the dependence on λ and k₀ but introduce an additional parameter analogous to λ for each active secondary pathway [26–29].

An expression for the evolution of the polymer number concentration, P(t) may be derived using Equation (9). Direct integration of Equation (2) gives the result for P(t):

Equations (9) and (10) give in closed form the time evolution of the mass and number concentrations of biofilaments growing through primary nucleation and filament elongation.

Insight into the early time behaviour of the polymer mass concentration can be obtained by expanding Equation (9) for early times to yield:

This expression recovers the characteristic ~t² dependence of the Oosawa theory and has an additional term linear in time relating to the growth of pre-formed aggregates.

In many cases, Equation (9) describes a sigmoidal function with a lag phase. The time of maximal growth rate, t_max, can be found from the inflection point of the sigmoid from the condition d²M/dt² = 0:

such that a lag phase exists only for:

Using the composition sinh ( artanh ( x ) ) = x / 1 - x 2 reduces this to the simple condition:

In other words, a point of inflection exists if the growth through elongation from the ends of pre-existing seeds, k₀, is less effective that the proliferation through nucleation and elongation of new material λ. This result implies that an increased nucleation rate promotes the existence of an inflection point, whereas an increased elongation rate or an increased level of seeding tends to disfavour its existence. In particular, we also note that in the absence of nucleation, an inflection point cannot exist in the polymer mass concentration as a function of time. This result is illustrated in Figure 1 where kinetic profiles are shown for increasing levels of pre-formed seed material added at the beginning of the reaction; a lag-phase is only observed when k₀/λ < 1, and the rate profile transforms from a sigmoidal shape at low pre-seeding levels to a concave form at high levels of seeding. Interestingly, the result Equation (14) is analogous to the criterion applicable for fragmentation dominated growth [26] where a lag phase exists only when the parameter controlling fragmentation-related secondary nucleation is larger than ek₀.

The maximal growth rate, r_max, is given by:

which occurs at a polymer mass concentration M_max given from Equation:

The lag time, τ_lag := t_max − M(t_max)/r_max, is then given by:

Interestingly, from Equation (17), we note that a point of inflection can never exist for P(t) for simple nucleated polymerisation. By contrast, when secondary pathways are active, an inflection point can frequently be present [27].

Many systems that evolve through nucleated polymerisation display characteristic scaling behaviour, including power-law relationships between phenomenological parameters, such as the lag-time and maximal growth rate, and the initial concentration of monomeric peptide [26–31]. This behaviour can be seen to be a consequence of the fact that under many conditions, the rate equations are dominated by a single parameter that corresponds to the dominant form of nucleation: λ for classical nucleated polymerisation through primary nucleation, Equation (9), or an analogous parameter [21,26,29] κ₋ or κ₂, respectively, for polymerisation in the presence of filament fragmentation of monomer-dependent secondary pathways [11,21,26,29,32–34], Table 1. These parameters have the general form 2 k + m ( 0 ) k N m ( 0 ) n where k_N = k_n; k₋; k₂ corresponds to the dominant nucleation process, k_n for primary nucleation, k₋ for filament fragmentation, k₂ for monomer-dependent secondary nucleation, and n is related to the monomer dependence of this process: n = n_c − 1, where n_c is the critical nucleus size for primary nucleation, n = 0 for fragmentation driven growth and n = n₂, the secondary nucleus size in cases where monomer-dependent secondary nucleation is dominant. The dominance of a single combination of the rate constants implies that many of the macroscopic system observables will be correlated since they are dependent on the same parameter. A striking examples of this behaviour is provided by the very general correlation between the lag-time and the maximal growth rate [29,30,35,36], which is manifested in the present case in Equations (15) and (17) as r_max ~ λ and τ_lag ~ λ⁻¹.

Interestingly the rate equations describe sigmoidal curves both in the presence and in the absence of secondary nucleation processes. For more complex primary nucleation pathways [19,21] the polynomial form for the early time solution is maintained, but higher-order terms are obtained. This observation implies that in the absence of secondary processes the lag-phase is, in general, less marked since the early time rise is a slower polynomial relationship rather than the exponential onset characteristic of secondary pathways [21]. However, the difference between a high-order polynomial and an exponential may not be apparent in experimental data in the presence of noise, and therefore a global analysis of the system under different conditions is required in order to obtain robust mechanistic information. This might involve, for example [29], experimental measurements at varying initial concentrations of the monomeric peptide; a global analysis of the measured reaction profiles in terms of integrated rate laws and the related scaling behaviours can then be used to establish information about the microscopic mechanisms of proliferation [29].