Growth and Dissolution of Macromolecular Markov Chains

Abstract

The kinetics and thermodynamics of free living copolymerization are studied for processes with rates depending on k monomeric units of the macromolecular chain behind the unit that is attached or detached. In this case, the sequence of monomeric units in the growing copolymer is a kth-order Markov chain. In the regime of steady growth, the statistical properties of the sequence are determined analytically in terms of the attachment and detachment rates. In this way, the mean growth velocity as well as the thermodynamic entropy production and the sequence disorder can be calculated systematically. These different properties are also investigated in the regime of depolymerization where the macromolecular chain is dissolved by the surrounding solution. In this regime, the entropy production is shown to satisfy Landauer’s principle.

Appendices

Appendix 1: Solving the Kinetic Equations

1.1 The Growth of 1st-order Markov Chains

The general method described in Sect. 2 to solve the master equations (5) have already been used in Ref. [22] to deduce Eqs. (41)–(43) with $k=1$. In this case, the attachment and detachment rates only depend on the previously incorporated monomeric units $w_{\pm m_l\vert m_{l-1}}$ and first-order Markov chains are generated.

1.2 The Growth of 2nd-order Markov Chains

Remarkably, the method of Ref. [22] extends mutatis mutandis to higher orders k. Since the equations become quite complicated, we start developing the method to the case $k=2$ where the attachment and detachment rates depend on the last two monomeric units $w_{\pm m_l\vert m_{l-1}m_{l-2}}$. Summing the master equations (5) over sequences $m_1m_2\ldots m_{l-r}$ with $r=0,1,2,\ldots$, we obtain a hierarchy of equations for the probabilities (7), (8), (9),.... This hierarchy is summarized with Eq. (14). Replacing with the particular solution and letting the wavenumber going to zero, the hierarchy reduces to Eq. (20). This hierarchy is divided by the constant $\phi _0(\emptyset )$ to get the equations

$$\begin{aligned} ({\varvec{\mathsf A}}+{\varvec{\mathsf B}}-{\varvec{\mathsf C}})\cdot \pmb {\mu }=0 \end{aligned}$$

(94)

for the time-independent probabilities (22), (23), (24), etc. The first few equations of this hierarchy read:

$$\begin{aligned} r=1:\quad 0= & {} \sum _{m_{l-2}m_{l-1}} w_{+m_l\vert m_{l-1}m_{l-2}}\, \mu (m_{l-2}m_{l-1}) \nonumber \\&+\, \sum _{m_{l-1}m_{l+1}} w_{-m_{l+1}\vert m_{l}m_{l-1}}\, \mu (m_{l-1}m_{l}m_{l+1}) \nonumber \\&- \sum _{m_{l-2}m_{l-1}} w_{-m_l\vert m_{l-1}m_{l-2}}\, \mu (m_{l-2}m_{l-1}m_{l}) \nonumber \\&- \sum _{m_{l-1}m_{l+1}} w_{+m_{l+1}\vert m_{l}m_{l-1}}\, \mu (m_{l-1}m_{l}), \end{aligned}$$

(95)

$$\begin{aligned} r=2:\quad 0= & {} \sum _{m_{l-2}} w_{+m_l\vert m_{l-1}m_{l-2}}\, \mu (m_{l-2}m_{l-1}) + \sum _{m_{l+1}} w_{-m_{l+1}\vert m_{l}m_{l-1}}\, \mu (m_{l-1}m_{l}m_{l+1}) \nonumber \\&- \sum _{m_{l-2}} w_{-m_l\vert m_{l-1}m_{l-2}}\, \mu (m_{l-2}m_{l-1}m_{l}) - \sum _{m_{l+1}} w_{+m_{l+1}\vert m_{l}m_{l-1}}\, \mu (m_{l-1}m_{l}), \nonumber \\ \end{aligned}$$

(96)

$$\begin{aligned} r=3:\quad 0= & {} w_{+m_l\vert m_{l-1}m_{l-2}}\, \mu (m_{l-2}m_{l-1}) + \sum _{m_{l+1}} w_{-m_{l+1}\vert m_{l}m_{l-1}}\, \mu (m_{l-2}m_{l-1}m_{l}m_{l+1}) \nonumber \\&- \bigg (w_{-m_l\vert m_{l-1}m_{l-2}} + \sum _{m_{l+1}} w_{+m_{l+1}\vert m_{l}m_{l-1}}\bigg )\, \mu (m_{l-2}m_{l-1}m_{l}), \end{aligned}$$

(97)

$$\begin{aligned} r=4:\quad 0= & {} w_{+m_l\vert m_{l-1}m_{l-2}}\, \mu (m_{l-3}m_{l-2}m_{l-1}) \nonumber \\&+ \sum _{m_{l+1}} w_{-m_{l+1}\vert m_{l}m_{l-1}}\, \mu (m_{l-3}m_{l-2}m_{l-1}m_{l}m_{l+1}) \nonumber \\&- \bigg ( w_{-m_l\vert m_{l-1}m_{l-2}}+ \sum _{m_{l+1}} w_{+m_{l+1}\vert m_{l}m_{l-1}}\bigg )\, \mu (m_{l-3}m_{l-2}m_{l-1}m_{l}), \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \vdots \nonumber \end{aligned}$$

(98)

Because of Eqs. (26), (27),..., summing Eq. (98) over $m_{l-3}$ gives Eq. (97), summing Eq. (97) over $m_{l-2}$ gives Eq. (96), and summing Eq. (96) over $m_{l-1}$ gives Eq. (95). We notice that summing Eq. (95) over $m_{l}$ gives $0=0$ for $r=0$. The key observation is that the equations for $r=3,4,\ldots$ are all similar to each other and they can thus be solved with the assumption of factorization

$$\begin{aligned} \mu (m_1m_2m_3\ldots m_{l-2}m_{l-1}m_l)= & {} \mu (m_1\vert m_2 m_3) \, \mu (m_2\vert m_3 m_4)\nonumber \\&\ldots \mu (m_{l-2}\vert m_{l-1} m_{l}) \, \mu (m_{l-1}m_{l}), \end{aligned}$$

(99)

as for a 2nd-order Markov chain in terms of the $M^3$ conditional probabilities $\mu (m_{l-2}\vert m_{l-1} m_{l})$ and $M^2$ tip probabilities $\mu (m_{l-1}m_{l})$. Under this assumption, all the equations $r=3,4,\ldots$ of the hierarchy reduce to the equation for $r=3$. Therefore, the conditional and tip probabilities should satisfy the first three equations for $r=1,2,3$, which become:

$$\begin{aligned} r=1:\qquad 0= & {} \sum _{m_{l-2}m_{l-1}} w_{+m_l\vert m_{l-1}m_{l-2}}\, \mu (m_{l-2}m_{l-1}) \nonumber \\&+ \sum _{m_{l-1}m_{l+1}} w_{-m_{l+1}\vert m_{l}m_{l-1}}\, \mu (m_{l-1}\vert m_{l}m_{l+1})\, \mu (m_{l}m_{l+1}) \nonumber \\&- \sum _{m_{l-1}}\bigg [\sum _{m_{l-2}} w_{-m_l\vert m_{l-1}m_{l-2}}\, \mu (m_{l-2}\vert m_{l-1}m_{l})+ \sum _{m_{l+1}} w_{+m_{l+1}\vert m_{l}m_{l-1}}\bigg ] \nonumber \\&\qquad \times \, \mu (m_{l-1}m_{l}), \nonumber \\&\end{aligned}$$

(100)

$$\begin{aligned} r=2:\qquad 0= & {} \sum _{m_{l-2}} w_{+m_l\vert m_{l-1}m_{l-2}}\, \mu (m_{l-2}m_{l-1}) \nonumber \\&+ \sum _{m_{l+1}} w_{-m_{l+1}\vert m_{l}m_{l-1}}\, \mu (m_{l-1}\vert m_{l}m_{l+1}) \, \mu (m_{l}m_{l+1})\nonumber \\&- \,\bigg [\sum _{m_{l-2}} w_{-m_l\vert m_{l-1}m_{l-2}}\, \mu (m_{l-2}\vert m_{l-1}m_{l}) + \sum _{m_{l+1}} w_{+m_{l+1}\vert m_{l}m_{l-1}}\bigg ]\nonumber \\&\qquad \times \, \mu (m_{l-1}m_{l}), \end{aligned}$$

(101)

$$\begin{aligned} r=3:\qquad 0= & {} w_{+m_l\vert m_{l-1}m_{l-2}}\, \mu (m_{l-2}m_{l-1}) \nonumber \\&+ \,\mu (m_{l-2}\vert m_{l-1}m_{l})\Bigg [\sum _{m_{l+1}} w_{-m_{l+1}\vert m_{l}m_{l-1}}\, \mu (m_{l-1}\vert m_{l}m_{l+1})\, \mu (m_{l}m_{l+1}) \nonumber \\&- \,\bigg (w_{-m_l\vert m_{l-1}m_{l-2}}+ \sum _{m_{l+1}} w_{+m_{l+1}\vert m_{l}m_{l-1}}\bigg ) \mu (m_{l-1}m_{l})\Bigg ]. \end{aligned}$$

(102)

We see that Eq. (101) should determine the tip probabilities $\mu (m_{l-1}m_{l})$ and Eq. (102) the conditional probabilities $\mu (m_{l-2}\vert m_{l-1}m_{l})$, if decoupling could be achieved between these quantities. This is where the partial velocities (40) are introduced. In the case $k=2$, they read

$$\begin{aligned} v_{m_{l-2}m_{l-1}}\equiv & {} \sum _{m_l} w_{+m_l\vert m_{l-1}m_{l-2}}\nonumber \\&-\, \frac{1}{\mu (m_{l-2}m_{l-1})}\sum _{m_l} w_{-m_l\vert m_{l-1}m_{l-2}} \, \mu (m_{l-2}\vert m_{l-1}m_{l}) \, \mu (m_{l-1}m_{l}).\qquad \end{aligned}$$

(103)

We observe that the first and third terms in the bracket of Eq. (102) actually form the partial velocity $v_{m_{l-1}m_{l}}$ multiplied by $\mu (m_{l-1}m_{l})$. Hence, Eq. (102) gives the conditional probabilities as

$$\begin{aligned} \mu (m_{l-2}\vert m_{l-1} m_{l}) = \frac{w_{+m_l\vert m_{l-1}m_{l-2}}\, \mu (m_{l-2}m_{l-1})}{(w_{-m_l\vert m_{l-1}m_{l-2}}+v_{m_{l-1}m_{l}})\, \mu (m_{l-1}m_{l})}. \end{aligned}$$

(104)

Inserting this result into Eq. (103), we get the self-consistent equations (41) with $k=2$ for the partial velocities:

$$\begin{aligned} v_{m_{l-2} m_{l-1}} = \sum _{m_l} \frac{w_{+m_l\vert m_{l-1}m_{l-2}}\, v_{m_{l-1}m_{l}}}{w_{-m_l\vert m_{l-1}m_{l-2}}+v_{m_{l-1}m_{l}}}. \end{aligned}$$

(105)

Summing Eq. (104) over $m_{l-2}$ and using the normalization condition (37), we find the equations (42) with $k=2$ for the tip probabilities:

$$\begin{aligned} \mu (m_{l-1} m_{l}) = \sum _{m_{l-2}} \frac{w_{+m_l\vert m_{l-1}m_{l-2}}}{w_{-m_l\vert m_{l-1}m_{l-2}}+v_{m_{l-1}m_{l}}} \, \mu (m_{l-2}m_{l-1}). \end{aligned}$$

(106)

We can check that Eq. (101) transformed with Eqs. (103)–(104) also yields Eq. (106), while Eq. (100) transformed with Eq. (104) becomes the trivial equation $v=v$ with the mean growth velocity (39) here given by

$$\begin{aligned} v = \sum _{m_{l-2}m_{l-1}} v_{m_{l-2}m_{l-1}} \, \mu (m_{l-2}m_{l-1})= \sum _{m_{l-1}m_{l}} v_{m_{l-1}m_{l}} \, \mu (m_{l-1}m_{l}). \end{aligned}$$

(107)

Therefore, Eqs. (100) and (101) are satisfied once Eq. (102) is solved by Eqs. (104)-(106) in terms of the partial velocities (103), which play an essential role in providing a self-consistent solution.

1.3 The Growth of kth-order Markov Chains

The general case with attachment and detachment rates $w_{\pm m_l\vert m_{l-1}\ldots m_{l-k}}$, depending on k previously incorporated monomeric units, can be solved with the same method. Now, the equations of the hierarchy (94) are similar to each other for $r=k+1,k+2,\ldots$:

$$\begin{aligned} r=k+1:&\nonumber \\ 0= & {} w_{+m_l\vert m_{l-1}\ldots m_{l-k}} \, \mu (m_{l-k}\ldots m_{l-1}) \nonumber \\&+\,\sum _{m_{l+1}} w_{-m_{l+1}\vert m_{l}\ldots m_{l-k+1}} \, \mu (m_{l-k}\ldots m_{l-1}m_{l}m_{l+1}) \nonumber \\&-\bigg ( w_{-m_{l}\vert m_{l-1}\ldots m_{l-k}} + \sum _{m_{l+1}} w_{+m_{l+1}\vert m_{l}\ldots m_{l-k+1}}\bigg )\, \mu (m_{l-k}\ldots m_{l-1}m_{l}), \end{aligned}$$

(108)

$$\begin{aligned} r=k+2:&\nonumber \\ 0= & {} w_{+m_l\vert m_{l-1}\ldots m_{l-k}} \, \mu (m_{l-k-1}m_{l-k}\ldots m_{l-1}) \nonumber \\&+\sum _{m_{l+1}} w_{-m_{l+1}\vert m_{l}\ldots m_{l-k+1}} \, \mu (m_{l-k-1}m_{l-k}\ldots m_{l-1}m_{l}m_{l+1}) \nonumber \\&-\bigg ( w_{-m_{l}\vert m_{l-1}\ldots m_{l-k}} + \sum _{m_{l+1}} w_{+m_{l+1}\vert m_{l}\ldots m_{l-k+1}}\bigg )\, \mu (m_{l-k-1}m_{l-k}\ldots m_{l-1}m_{l}),\qquad \quad \,\, \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \vdots \nonumber \end{aligned}$$

(109)

so that they are solved with the assumption (36) that the Markov chain is of kth order. The equation (108) for $r=k+1$ becomes

$$\begin{aligned} r= & {} k+1:\nonumber \\&0= w_{+m_l\vert m_{l-1}\ldots m_{l-k}} \, \mu (m_{l-k}\ldots m_{l-1})+\mu (m_{l-k}\vert m_{l-k+1}\ldots m_{l}) \nonumber \\&\qquad \times \, \Bigg [\sum _{m_{l+1}} w_{-m_{l+1}\vert m_{l}\ldots m_{l-k+1}} \, \mu (m_{l-k+1}\vert m_{l-k+2} \ldots m_{l+1})\, \mu (m_{l-k+2}\ldots m_{l+1}) \nonumber \\&\qquad -\bigg ( w_{-m_{l}\vert m_{l-1}\ldots m_{l-k}} + \sum _{m_{l+1}} w_{+m_{l+1}\vert m_{l}\ldots m_{l-k+1}}\bigg )\, \mu (m_{l-k+1}\ldots m_{l}) \Bigg ]. \end{aligned}$$

(110)

Thanks to the introduction of the partial velocities (40), Eq. (110) gives the expression (43) for the conditional probabilities $\mu (m_{l-k}\vert m_{l-k+1}\ldots m_{l})$. On the one hand, the self-consistent equations (41) for the partial velocities are obtained by inserting Eq. (43) into the definition (40) for the partial velocities. On the other hand, Eq. (42) for the tip probabilities are given by summing Eq. (43) over $m_{l-k}$ and using the normalization conditions (37).

Here also, we can check that the equations for $r=1,2,\ldots ,k$ in the hierarchy are satisfied with the assumption (36) and Eqs. (40)–(43).

Appendix 2: Proof of the Fluctuation Relation

In order to prove the fluctuation relation (67), we should start from the master equation

$$\begin{aligned}&\frac{d}{dt}\, P_t(m_1\ldots m_l,\Delta \mathbf{N})\nonumber \\&\quad = w_{+m_l\vert m_{l-1}\ldots m_{l-k}} \, P_t(m_1\ldots m_{l-1},\Delta \mathbf{N}-\mathbf{1}_{m_{l-k}\ldots m_l}) \nonumber \\&\qquad +\,\sum _{m_{l+1}=1}^M w_{-m_{l+1}\vert m_{l}\ldots m_{l-k+1}} \, P_t(m_1\ldots m_{l+1},\Delta \mathbf{N}+\mathbf{1}_{m_{l-k+1}\ldots m_{l+1}}) \nonumber \\&\qquad -\, \left( w_{-m_{l}\vert m_{l-1}\ldots m_{l-k}} + \sum _{m_{l+1}=1}^M w_{+m_{l+1}\vert m_{l}\ldots m_{l-k+1}}\right) P_t(m_1\ldots m_{l},\Delta \mathbf{N}), \end{aligned}$$

(111)

for the probability $P_t(m_1\ldots m_l,\Delta \mathbf{N})$ that the chain has the sequence $m_1\ldots m_l$ with $l\gg k$ at the time t and that the $M^{k+1}$ counters of the different possible multiplets $m_j\ldots m_{j+k}$ of length $k+1$ take the values $\Delta \mathbf{N} = \left\{ \Delta N_{m_j\ldots m_{j+k}}\right\}$. The notation

$$\begin{aligned} (\Delta \mathbf{N}\pm \mathbf{1}_{m_{i}\ldots m_{i+k}})_{m_j\ldots m_{j+k}}= \left\{ \begin{array}{ll} \Delta N_{m_j\ldots m_{j+k}}, &{}\text{ if } \quad m_j\ldots m_{j+k}\ne m_{i}\ldots m_{i+k} \\ \Delta N_{m_j\ldots m_{j+k}}\pm 1, &{}\text{ if } \quad m_j\ldots m_{j+k}=m_{i}\ldots m_{i+k} \end{array} \right. \qquad \end{aligned}$$

(112)

is used in Eq. (111).

On the one hand, we notice that the master equation (5) is recovered from Eq. (111) by summing over $\Delta \mathbf{N}$. On the other hand, the probability distribution appearing in the fluctuation relation (67) is defined as

$$\begin{aligned} p_t(\Delta \mathbf{N}) \equiv \sum _{m_1\ldots m_l} P_t(m_1\ldots m_l,\Delta \mathbf{N}). \end{aligned}$$

(113)

In the long-time limit, the same factorization as in Eq. (35) is expected:

$$\begin{aligned} P_t(m_1\ldots m_{l-1}m_l,\Delta \mathbf{N})\simeq p_t(\Delta \mathbf{N}) \, \mu (m_1 \ldots m_{l-1} m_l). \end{aligned}$$

(114)

Therefore, the probability distribution (113) evolves in time according to

$$\begin{aligned} \frac{dp_t}{dt} = \hat{L} \, p_t \end{aligned}$$

(115)

with the linear operator

$$\begin{aligned} \hat{L}\equiv & {} \sum _{m_{l-k}\ldots m_l} \left[ w_{+m_l\vert m_{l-1}\ldots m_{l-k}} \, \mu (m_{l-k}\ldots m_{l-1}) \, \left( \hat{E}^{-}_{m_{l-k}\ldots m_l}-1\right) \right. \nonumber \\&\left. +\, w_{-m_l\vert m_{l-1}\ldots m_{l-k}} \, \mu (m_{l-k}\vert m_{l-k+1}\ldots m_l) \, \mu (m_{l-k+1}\ldots m_l) \, \left( \hat{E}^{+}_{m_{l-k}\ldots m_l}-1\right) \right] \nonumber \\ \end{aligned}$$

(116)

where

$$\begin{aligned} {\hat{E}}^{\pm }_{m_{l-k}\ldots m_l} f(\Delta \mathbf{N}) = f(\Delta \mathbf{N}\pm \mathbf{1}_{m_{l-k}\ldots m_l}). \end{aligned}$$

(117)

The cumulant generating function of the counters $\Delta \mathbf{N}$ is defined as

$$\begin{aligned} Q(\pmb {\lambda }) \equiv \lim _{t\rightarrow \infty } -\frac{1}{t} \ln \left\langle \mathrm{e}^{-\pmb {\lambda }\cdot \Delta \mathbf{N}}\right\rangle _t \end{aligned}$$

(118)

in terms of the counting parameters $\pmb {\lambda }=\{\lambda _{m_{l-k}\ldots m_l}\}\in {\mathbb R}^{M^{k+1}}$. The cumulant generating function can be obtained as the leading eigenvalue, $\hat{L}_{\pmb {\lambda }}\,\varphi = -Q(\pmb {\lambda })\,\varphi$, for the modified operator

$$\begin{aligned} \hat{L}_{\pmb {\lambda }} \equiv \mathrm{e}^{-\pmb {\lambda }\cdot \Delta \mathbf{N}} \hat{L} \, \mathrm{e}^{+\pmb {\lambda }\cdot \Delta \mathbf{N}}, \end{aligned}$$

(119)

giving

$$\begin{aligned} Q(\pmb {\lambda })= & {} \sum _{m_{l-k}\ldots m_l} \Big [ w_{+m_l\vert m_{l-1}\ldots m_{l-k}} \, \mu (m_{l-k}\ldots m_{l-1}) \, \left( 1-\mathrm{e}^{-\lambda _{m_{l-k}\ldots m_l}}\right) \nonumber \\&+\, w_{-m_l\vert m_{l-1}\ldots m_{l-k}} \, \mu (m_{l-k}\vert m_{l-k+1}\ldots m_l) \, \mu (m_{l-k+1}\ldots m_l) \, \left( 1-\mathrm{e}^{+\lambda _{m_{l-k}\ldots m_l}}\right) \Big ].\nonumber \\ \end{aligned}$$

(120)

Since the sum extends over all the multiplets of length $k+1$, the indices can be arbitrarily changed as $m_{l-k}\ldots m_l \rightarrow m_j\ldots m_{j+k}$ for any integer j.

The net rates (52) or (54) are obtained by differentiating the cumulant generating function (120) with respect to the counting parameters:

$$\begin{aligned} \mathbf{J} = \frac{\partial Q}{\partial \pmb {\lambda }}\Big \vert _{\pmb {\lambda }=0} = \lim _{t\rightarrow \infty } \frac{1}{t} \langle \Delta \mathbf{N}\rangle _t. \end{aligned}$$

(121)

The remarkable property is that the cumulant generating function (120) obeys the symmetry relation

$$\begin{aligned} Q(\pmb {\lambda }) =Q(\mathbf{A}-\pmb {\lambda }), \end{aligned}$$

(122)

with respect to the affinities (53). Using large-deviation theory [46, 47], the fluctuation relation (67) is thus established.