Molecular species and reaction parameters

We assume that the dynamics of protein heterodimer formation by post- and co-translational assembly depend on the following system parameters [22], see also Fig 1: The synthesis of nascent peptide chains and requires encountering and binding of corresponding mRNA molecules and ribosomes. In a cell, the distributions of mRNAs as well as ribosomes are non-uniform and time-dependent. Therefore, nascent chain synthesis is initiated with time- and position-dependent rates and , respectively. Nascent chains are tethered to the translating ribosomes and, thus, immobilized until translation has terminated with constant rates and , respectively. After translation has terminated, the nascent chains are released from the ribosomes and become free protein subunits and , respectively. A free subunit can bind to a nascent chain with binding rate constant κ_co. The formed complex is immobile until the synthesis of subunit is finished and the complex is released from the ribosome with rate . The released complex is called co-translationally assembled complex to reflect its formation process. Alternatively, a free subunit binds a free (released) subunit with binding rate constant κ_post to form a post-translationally assembled complex . Free subunits as well as co- and post-translationally assembled complexes diffuse with diffusion constants , , and , respectively. With these system parameters, the time evolution of the concentrations of nascent chains and , protein subunits and , nascent complex and mature complexes and is described by (1a) (1b) (1c) (1d) (1e) (1f) (1g) As usual, Δ denotes the Laplacian with respect to the spatial variable only.

Remark 1. Some first observations show that and remain bounded as long as and are bounded functions. Furthermore, (2)

For the sake of clarity, we make two simplifications to this reaction-diffusion system.

Solvability

In this brief section we will give a basic result on the full system (4). Since the main focus of the analytical investigations in this work will lie on the special case of d_a = d_b = 0, we keep the proof to a short outline.

Theorem 3. We assume (5), (6), (7) and (10).

Then there is a unique global solution of (4), (9), (8), i.e. a triplet of functions such that and (4), (9), (8) are satisfied at each point.

This solution moreover satisfies for all (x, t) ∈ Ω × (0, ∞).

Proof. The estimates can be obtained from comparison arguments. If d_a = d_b = 0, (4) is a system of ordinary differential equations (ODEs), and existence and uniqueness of a local solution are asserted by the Picard–Lindelöf theorem. That this solution is global follows from the bounds given above. For positive d_a and d_b, we prove existence by employing a Schauder fixed point reasoning, which relies on general parabolic regularity (mainly [23, Theorems 14.4, 14.6, 15.5], [24, Theorem 4]) and (for Hölder regularity in the coupled PDE-ODE system) on a result like [25, Lemma 2.1]. Uniqueness is easily derived with the help of Grönwall’s inequality.

The homogeneous case: d_a = d_b = 0, a₀, b₀, n₀, κ_a, κ_b constant

In this section, we investigate the system in the spatially homogeneous setting, finally giving a complete characterization of the long-term behavior of solutions with respect to the relative importance of the reaction pathways.

In this simpler scenario, (4) is reduced to the ODE system (11a) (11b) (11c) according to (9) and (10) supplemented with initial conditions (11d)

A first general observation, irrespective of the size of the involved parameters, is the following conserved quantity: (12) where . Note that this corresponds to (3) for (1).

The case of overproduction of : κ_a < κ_b

If there is an overproduction of , the post-translational assembly pathway dominates: , more precisely:

Lemma 4. Let d_a = d_b = 0 and let a₀, b₀, n₀, κ_a, κ_b, γ be positive constants with κ_a < κ_b. Then the solution to (11) satisfies Proof. According to (12), a − (n + b) → −∞ as t → ∞, which, due to a ≥ 0, implies n + b → ∞. Given any M > 0, there is T > 0 such that for all t > T we have n(t) + γb(t) > M and thus which shows that . As M was arbitrary and a ≥ 0, therefore lim_t→∞ a(t) = 0.

For every ε > 0, one can find T > 0 such that for t > T, a(t) < ε. For such T, showing that . As, additionally, lim sup_t→∞ n(t) ≤ κ_b (because n_t ≤ κ_b − n on (0, ∞)), we obtain n(t) → κ_b as t → ∞.

As n(t) + b(t) → ∞ as t → ∞, this means that b(t) → ∞ as t → ∞.

Subsequently, can also conclude from (11) that n_t → 0 and a_t − b_t → κ_a − κ_b.

The case of overproduction of : κ_b < κ_a

If there is an overproduction of , the concentrations of both and vanish in the large-time limit, as both are immediately used in reactions.

Lemma 5. Let d_a = d_b = 0 and let a₀, b₀, n₀, κ_a, κ_b, γ be positive constants with κ_a > κ_b. Then the solution to (11) satisfies (13) Proof. By (12), a ≥ a − (n + b) → ∞ as t → ∞; in particular, a(t) → ∞ as t → ∞.

Let M > 0. Then there is T > 0 such that a(t) > M for all t > T. On (T, ∞), we have Therefore, by a comparison argument, Employing this reasoning for arbitrarily large M, we obtain that lim_t→∞ n(t) = 0.

Given ε > 0 and M > 0, there is T > 0 such that for every t > T we have n(t) < ε and . Hence, on (T, ∞), so that , i.e. b(t) → 0 as t → ∞.

Remark 6. According to Lemma 4 and Lemma 5, in both cases κ_a < κ_b and κ_a > κ_b, the trajectories of the ODE system (11) are not persistent (cf. [12, Def. 2.12]).

Although both concentrations n and b tend to 0, we can still reasonably ask which of the reaction pathways is stronger, that is how the quotient behaves. Even for large γ—i.e. when the binding rate constant for post-translational assembly exceeds that for co-translational assembly—it is almost immediately obtained from a study of that the co-translational pathway wins over the post-translational. As we will find in Lemma 8, for small γ > 0 the result is the same, although it is not as easily seen from the system.

Lemma 7. In addition to the assumptions of Lemma 5 let γ ≥ 1. Then Proof. In order to see this, we introduce and show that w(t) → 0 as t → ∞. We conclude from (11) that (14) Since γ ≥ 1, the term (1 − γ)aw(1 − w) is negative so that (14) shows Due to (κ_b > 0 and) n + b → 0 (Lemma 5), given M > 0 we find T > 0 such that on (T, ∞) we have , that is i.e. , hence lim_t→∞ w(t) = 0.

Lemma 8. In addition to the assumptions of Lemma 5 let γ < 1. Then Proof. We show this in two steps: Firstly, ab → 0 as t → ∞ (Lemma 9), secondly, an → κ_b as t → ∞ (Lemma 10), so that as t → ∞.

Lemma 9. Under the assumptions of Lemma 8, a(t)b(t) → 0 as t → ∞.

Proof. Concerning the evolution of ab, system (11) implies Let us assume that lim sup_t→∞ (ab)(t) ≥ δ for some δ > 0. Relying on (13) we choose t₁ > 0 such that and note that (15)

Furthermore, we let be an increasing sequence with limit ∞ such that for each , introduce and assume that M_k ≠ ∅. Then is well-defined and According to (15), for these t, so that a contradiction. Hence, M_k = ∅, that is Again by (15), we therefore may conclude that on [t₁, ∞), which implies (ab)(t) → −∞ as t → ∞, in contradiction to the nonnegativity of ab.

We conclude that lim sup_t→∞ (ab)(t) = 0 and thus ab → 0.

Similar reasoning shows an → κ_b:

Lemma 10. Under the assumptions of Lemma 8, a(t)n(t) → κ_b as t → ∞.

Proof. We assume that lim sup_t→∞ an ≥ κ_b + δ for some δ > 0 and, aided by (13), let t₁ > 0 be such that Letting be a monotone sequence with such that we let and . If we assume that M_k ≠ ∅, then exists and satisfies and for , we have according to the definition of . This implies that on , in particular , contradicting the definitions of and . Therefore, M_k = ∅ for each and As above, this entails that on (t₁, ∞), which in turn proves an → −∞, in contradiction to the nonnegativity of a and n. In conclusion, lim sup_t→∞ (an) ≤ κ_b.

Now we assume liminf_t→∞ (an) ≤ κ_b − δ for some δ > 0. With t₁ > 0 chosen such that a > 1 and on (t₁, ∞), we have If is, again, a monotone increasing divergent sequence such that for every , and—under the assumption that M_k be nonempty—, we see that on , and thus on . As consequence, , contradicting the definitions of and . Thus, M_k = ∅ and Therefore on (t₁, ∞), so that (an)(t) → ∞ as t → ∞, which contradicts lim sup_t→∞ (an)(t) ≤ κ_b as well as the assumption liminf_t→∞ an ≤ κ_b − δ. In conclusion, liminf_t→∞ (an)(t) ≥ κ_b. Together with the first part, this shows lim_t→∞ (an)(t) = κ_b.

The special case of balanced production κ_a = κ_b

In the previous two subsections we have seen that if the production of either or exceeds that of the other, this component accumulates in the system and determines which of the reaction pathways is more important on long time scales. We will now, in contrast, consider the case where the production rates of and are precisely in balance: κ_a = κ_b.

While it can be argued that exact equality of parameters is never found in reality, this case is interesting as the critical case where the system behavior is not determined by oversaturation with one of the two proteins. (Taking into account that the assumption of time-independence of the parameters already is an approximation that hides fluctuations, equality of these parameters can on the other hand be seen as the relevant and most appropriate choice among constants for all scenarios where there is no unlimited buildup of any of the two components in the long term.)

In this case we set and first observe that any surplus of one of the protein types is conserved for all times: (16) This allows us to write (11) equivalently as (17a) (17b) or (18a) (18b) or (19a) (19b) We can already note a first difference to the earlier cases where one of the concentrations grew without bounds:

Lemma 11. Let d_a = d_b = 0 and let a₀, b₀, n₀, γ be positive constants and κ = κ_a = κ_b > 0. Then there are constants such that the solution (a, n, b) to (11) satisfies for all t ∈ (0, ∞). Moreover, (20) with c₀ as in (16).

Proof. From (17b) boundedness of n from above is immediate and subsequently (19b) and (18a) make boundedness of b and a, respectively, obvious. Using boundedness of a and (17b), we also find a positive lower bound for n; (18a) and entail a lower bound of a, whereas and (19b) yield a positive lower bound for b.

As to (20), we know from (11), nonnegativity of n and b and (16) that Since the solution of satisfies , where we conclude (from a comparison argument) that .

Lemma 11 shows that the relative importance of the reaction pathways remains bounded between positive constants.

Lemma 12. Let d_a = d_b = 0 and let a₀, b₀, n₀, γ be positive constants and κ = κ_a = κ_b > 0. If the solution (a, n, b) of (11) converges as t → ∞, then where a_∞ is the unique positive solution of (21) with c₀ as in (16).

Proof. The only possible limits for convergent solutions of ODEs are the steady states. Given , all steady states fulfilling (16) are characterized by the equations given in this lemma. Among the roots of p exactly one is positive, and according to Lemma 11 this is the only solution of (21) that could be a limit of a.

Theorem 13. Let d_a = d_b = 0 and let a₀, b₀, n₀, γ be positive constants and κ = κ_a = κ_b > 0. Then the solution (a, n, b) of (11) converges, and with a_∞ being the root of (21), which is monotone increasing with respect to κ and c₀ = a₀ − n₀ − b₀ and decreasing with respect to γ.

Proof. All possible limits of convergent solutions have been identified in Lemma 12. It remains to show that all solutions actually converge. We treat different ranges of values of γ and c₀ separately.

Case I: If γ ≤ 1, (17) is a competitive (two-dimensional) system. All of its bounded solutions (hence, by Lemma 17: all solutions) therefore converge, see [26].

Case II: γ > 1, : We cover this case with the following Lyapunov type reasoning: Starting from (18), for arbitrary B > 0 we compute where a_∞ is taken from Lemma 12 and so that 0 = −b_∞ − c₀ + a_∞ − γa_∞b_∞ and . We note that implies and hence is positive and satisfies (1 − γ)a_∞ + B − Bγb_∞ = 0.

By Lemma 11, there are T > 0 and δ > 0 such that 2(a_∞ + a(t) − c₀ + (γ − 1)b(t)) ≥ δ and for all t ≥ T. Therefore, satisfies showing that a(t) → a_∞ and b(t) → b_∞ as t → ∞.

Case IIIa: γ > 1, : We assume there is some t₀ ≥ 0 such that . We note that is a subsolution to (19b): because . Therefore, by a comparison argument, for all t > t₀. On the set , system (19) is competitive, and hence its solutions converge according to Hirsch’s result on two-dimensional cooperative and competitive systems of ODEs [26].

Case IIIb: γ > 1, : Now we assume for all t > 0. In this case, according to (18b), and therefore If , this contradicts the assumption for all times; if , then and convergence of n or a is easily obtained from (18a) or (19a).