We revisit the classical problem of nucleated polymerisation and derive a range of exact results describing polymerisation in systems intermediate between the well-known limiting cases of a reaction starting from purely soluble material and for a reaction where no new growth nuclei are formed.
The classical theory of nucleated polymerisation [1] describes the growth of filamentous structures formed through homogeneous nucleation [2][3][4][5][6][7]. This framework was initially developed by Oosawa and coworkers in the 1960s [1,8] to describe the formation of biofilaments, including actin and tubulin. This theory has been generalised to include secondary nucleation processes by Eaton and Ferrone [9] in the context of their pioneering work elucidating the polymerisation of sickle haemoglobin, and by Wegner [10] in order to include fragmentation processes into the growth model for actin filaments.
For irreversible growth in the absence of pre-formed seed material and secondary nucleation pathways, in 1962 Oosawa presented solutions to the kinetic equations which were very successful in describing a variety of characteristics of the polymerisation of actin and tubulin. The other limiting case, namely where seed material is added at the beginning of the reaction and where no new growth nuclei are formed during the reaction, is also well known. In this paper, we present exact results which encompass all cases between these limiting scenarios, extending the results of Oosawa for a system dominated by primary nucleation to the case where an arbitrary concentration of pre-formed seed material is present.
We also discuss a range of general closed form results from the Oosawa theory for the behaviour of a system of biofilaments growing through primary nucleation and elongation. We then compare the behaviour of systems dominated by primary nucleation to results derived recently for systems dominated by secondary nucleation.
A. Derivation of the rate laws for the polymer number and mass concentrations
The theoretical description of the polymerisation of proteins such as actin and tubulin to yield functional biostructures was considered in the 1960s by Oosawa [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 Eq. ( 1) originates from the assumption of growth from both ends. For the case of irreversible biofilament growth, the polymerisation rate dominates over the depolymerisation rate; from Eq. ( 1), the rate of change of the number of filaments, P (t), and the free monomer concentration, m(t), were shown by Oosawa under these conditions [1,8] to obey:
Combining Eqs. ( 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 Eqs. ( 2) and ( 3), the substitution z(t) := log(m(t))
followed by multiplication through by dz/dt yields:
Integrating both sides results in:
we obtain a separable equation for dz/dt, which can be solved to yield:
Inserting the appropriate boundary conditions in terms of m(0) and P (0) fixes the values of the constants A and B, resulting in the final exact result for the polymer mass concentration
where the effective rate constant λ is given by λ
We note that this expression only depends on two combinations of the microscopic rate constants, k 0 = 2k + P (0) and λ. The result reveals that λ controls the aggregation resulting from the newly formed aggregates, whereas k 0 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 Eq. ( 9) yields the result presented by Oosawa [1] and the single relevant parameter in the rate equations is λ. Interestingly, generalisations of Eq. ( 9) which include secondary pathways, maintain the dependence on λ and k 0 but introduce an additional parameter analogous to λ for each active secondary pathway [11][12][13][14].
An expression for the evolution of the polymer number concentration, P (t) may be derived using Eq. ( 9). Direct integration of Eq. ( 2) gives the result for P (t):
Eqs. ( 10) and ( 9) give in closed form the time evolution of the biofilament number and mass concentration growing through primary nucleation and filament elongation.
Insight into the early time behaviour of the polymer mass concentration can be obtained by expanding Eq. ( 9) for early times to yield: the exact solution to the rate equations Eq. ( 9); the thin solid lines are calculated from numerical simulations of the master equati
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