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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 [

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 theoretical description of the polymerisation of proteins such as actin and tubulin to yield functional biostructures was considered in the 1960s [

where _{+}, _{off}, _{n}_{c}_{c}

For the case of irreversible biofilament growth, the polymerisation rate dominates over the depolymerisation rate; from

Combining

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

Integrating both sides results in:

This is a separable equation for

Integration and exponentiation yields the expression for

The values of the constants _{tot} − _{tot} being the total monomer concentration. This results in the exact integrated rate law:

where the effective rate constant _{c}

We note that this expression only depends on two combinations of the microscopic rate constants, _{0} = 2_{+}_{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, _{tot}, these expressions reduce to _{0} but introduce an additional parameter analogous to

An expression for the evolution of the polymer number concentration,

Insight into the early time behaviour of the polymer mass concentration can be obtained by expanding

This expression recovers the characteristic ~^{2} dependence of the Oosawa theory and has an additional term linear in time relating to the growth of pre-formed aggregates.

In many cases, _{max}, can be found from the inflection point of the sigmoid from the condition ^{2}^{2} = 0:

such that a lag phase exists only for:

Using the composition

In other words, a point of inflection exists if the growth through elongation from the ends of pre-existing seeds, _{0}, is less effective that the proliferation through nucleation and elongation of new material _{0}/_{0}.

The maximal growth rate, _{max}, is given by:

which occurs at a polymer mass concentration _{max} given from Equation:

The lag time, _{lag} := _{max} − _{max})/_{max}, is then given by:

Interestingly, from

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 [_{−} or _{2}, respectively, for polymerisation in the presence of filament fragmentation of monomer-dependent secondary pathways [_{N}_{n}_{−}_{2} corresponds to the dominant nucleation process, _{n}_{−} for filament fragmentation, _{2} for monomer-dependent secondary nucleation, and _{c}_{c}_{2}, 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 [_{max} ~ _{lag} ~ ^{−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 [

In this paper, we have provided results for the time course of nucleated polymerisation for systems that are initially in a mixed state and contain both monomeric and fibrillar material. These results generalise the classical Oosawa theory that describes the formation of biofilaments to cases where an arbitrary amount of pre-formed seed material is present in the system. Furthermore, these results represent a reference to which polymerisation driven by secondary pathways can be compared.

We are grateful to the Schiff Foundation (SIAC), and to the Wellcome (MV, CMD, TPJK) and Leverhulme Trusts (CMD) for financial support.

Nucleated polymerisation in the presence of seed material. The thick dashed lines are the exact solution to the rate equations _{t}_{=0} = _{+}_{0}/_{tot} = 10 _{c}_{n}_{tot}^{nc−1} = 1 · 10^{−9} s^{−1}, _{+} = 1 · 10^{5} M^{−1} s^{−1}.

Comparison of biofilament growth dominated by primary and secondary nucleation pathways. Primary nucleation processes create new aggregates at a rate that depends only on the concentration of monomeric peptide, whereas fragmentation creates new aggregates at a rate that depends only on the concentration of existing aggregates; monomer-dependent secondary nucleation creates new aggregates at a rate that depends on both the concentration of monomeric peptide and the concentration of existing aggregates. The dependencies of the latter two (secondary) nucleation processes on the existing aggregate concentration results in positive feedback: as the reaction proceeds, and proliferation through these mechanisms increases the concentration of aggregates, the rate at which these processes occur further is increased.

Primary nucleation | Fragmentation | Monomer-dependent secondary nucleation | |
---|---|---|---|

_{+} |
_{−}, _{+} |
_{2}, _{+} | |

Polynomial | Exponential | Exponential | |

Yes with |
Yes with _{−} |
Yes with _{2} | |

No | Yes | Yes |