---
abstract: 'A numerical simulation shows that the osmotic pressure of compressed lattice knots is a function of knot type, and so of entanglements.  The osmotic pressure for the unknot goes through a negative minimum at low concentrations, but in the case of non-trivial knot types $3_1$ and $4_1$ it is negative for low concentrations.  At high concentrations the osmotic pressure is divergent, as predicted by Flory-Huggins theory.  These results suggest that the physical and chemical properties of biopolymers in confined spaces will be a function of entanglements.' 
address: 'Department of Mathematics & Statistics, York University, Toronto, ON, Canada, M3J 1P3'
author:
- EJ Janse van Rensburg
title: Osmotic pressure of compressed lattice knots
---

Compressed lattice knots
========================

It is known that entanglements have an effect on the physical properties of biopolymer such as DNA [@Cozzarelli; @Rybenkov].  Topological entanglement (knotting and linking) also impedes the movement of DNA, for example in the ejection of DNA from a viral capsid [@Marenduzzo].  Confinement of a biopolymer (eg.  in an organel in a living cell) causes an increase in knotting (and so in topological entanglement) [@Orlandini], and also compresses the biopolymer.  This compression is known to induce more entanglements [@Tang].  Biopolymers are also assembled and disassembled in confining environments in the cell by the addition or removal of monomers.  In equilibrium the tendency of a polymer to grow or evaporate can be examined by calculating the osmotic pressure of monomers, and in this letter it is shown that this is a function of knotting, and so of the level of entanglements. 

A polymer in a confined space in three dimensions can be modelled by a random string [@Flory56; @Edwards65; @Edwards67; @Edwards68] confined to a cubical volume or cavity.  If the random string has circular topology, then entanglements can be captured by knotting the string, as shown in Fig.  \[figure1\]. 

The entropy of a random string can be quantified by putting it on a lattice to get a lattice self-avoiding walk model of polymers [@Flory56; @Hammersley57; @Flory69].  If the self-avoiding walk is circular, then it is a model of a ring polymer called a *lattice polygon* [@Hammersley61; @Frisch61; @Delbruck62].  Lattice polygons are knotted asymptotically with probability $1$ [@Sumners88; @Pippenger89].  A lattice polygon with fixed knot type is a *lattice knot* [@DeGennes84; @Diao1993].  It is known that the entropy of a lattice knot is a function of its knot type [@Orlandini1998; @JvR2014]. * Tight lattice knots* [@DeGennes84] are minimal length lattice knots [@JvR1995; @JvR2011; @Scharein2009].  The compressibility of tight lattice knots is a function of knot type [@JvR2012; @Gasumova2012]. 

In Fig.  \[figure2\] the model in Fig.  \[figure1\] is quantified by a placing and compressing a lattice knot in a cubical box.  The entropy of the *compressed lattice knot* has contributions from translational degrees of freedom (if it is small compared to the side length of the box), from topological constraints (due to entanglements which depends on the knot type), and from conformational degrees of freedom.  In this paper compressed lattice knots of three knot types [@Rolfsen], namely the *unknot* ($0_1$), the *trefoil* ($3_1$), and the *figure eight knot* ($4_1$) will be considered; see Fig.  \[figure3\]. 

A cube in the lattice of side-length $L{-}1$ has *volume* $V=L^3$ and *dimension* $L$. The maximum length of a lattice knot confined to a cube of dimension $L$ is $L^3$ if $L$ is even, and $L^3{-}1$ if $L$ is odd.  The *lattice unknot* has minimal length $4$ and there are $3L(L{-}1)^2$ ways it can be placed in the cube.  The *lattice trefoil knot* $3_1$ can be tied with $24$ steps in the cubic lattice [@Diao1993], and there are $3328$ conformations distinct under translations in the cubic lattice [@Scharein2012].  None of these tight lattice trefoils can be realised in a cube of dimension $3$, but a numerical simulation detected $4168$ distinct placements of $3304$ tight lattice trefoils in a cube of dimension $4$, and $30104$ distinct placements of tight lattice trefoils in a cube of dimension $5$. Similarly, a *tight lattice figure eight knot* $4_1$ has minimal length $30$ in the cubic lattice [@Scharein2012] and there are $3648$ conformations distinct under translations in the cubic lattice.  A computer count shows that none of these can be realised in a cube of dimension $3$, but there are $864$ distinct placements of tight lattice figure eight knots in a cube of dimension $4$, and $18048$ distinct placements in a cube of dimension $5$.

Free energy and osmotic pressure
================================

Denote the number of distinct placements of lattice knots of length $n$, of knot type $K$, confined in a cube of dimension $L$, by $p_{n,L}(K)$. Then, for example, $p_{24,3}(3_1)=0$ and $p_{24,4}(3_1)=4168$. Approximate enumeration of $p_{n,L}(K)$ can be done by using the GAS algorithm [@JvR2009; @JvR2011A] implemented with BFACF moves [@BFACF1; @BFACF2]. 

The concentration of vertices in a lattice knot in a cube of dimension $L$ is $\phi=\sfrac{n}{V}$ where $V=L^3$. The free energy at concentration $\phi$ of lattice knots of type $K$ is $$F_{tot} (\phi;K) = - \log p_{n,L} (K) ,$$ where $n=\phi\,V$. The free energy per unit volume is $F_L(\phi;K) = \sfrac{1}{V} F_{tot} (\phi;K)$ and this is plotted in Fig.  \[figure4\] for $2\leq L \leq 15$ and for $K=0_1$ (the unknot) against the monomer concentration $\phi$. The shape of these curves is consistent with prediction of Flory-Huggins theory [@DeGennes79]. 

The osmotic pressure $\Pi(\phi;K)$ of compressed lattice knots is given by $$\Pi(\phi;K) = - \Sfrac{d}{dV}\, F_{tot}(\phi;K) .$$ Changing variables to $\phi$ shows that $$\Pi(\phi;K) = \phi^2 \, \Sfrac{d}{d\phi} \LB \sfrac{1}{\phi} \, F_L(\phi;K) \RB$$ in terms of the free energy per unit volume.  This can be computed from the data in figure \[figure4\] by taking a numerical derivative.  Using a central second order numerical approximation to the derivative gives Fig.  \[figure5\].  This appears to be consistent with the predicted Flory-Huggins osmotic pressure: $\Pi(\phi;0_1)$ is increasing and sharply diverges as $\phi\to 1^-$. Closer examination of Fig.  \[figure5\] shows that $\Pi(\phi;0_1)$ is not monotone but is decreasing a low concentrations and negative and not monotonic on an interval of low concentrations; see Fig.  \[figure6\]. 

At negative osmotic pressure the lattice unknot will add length.  Similarly, at positive osmotic pressure the lattice unknot will shed length and become smaller.  The pressure curves in Fig.  \[figure6\] are functions of $L$ and each has two zeros at $\phi_0$ and $\phi_m$. At concentrations $\phi<\phi_0$ the lattice unknot will evaporate, and when $\phi_0<\phi<\phi_m$ it will add length until the concentration is $\phi_m$ (which is a stable fixed point).  It will also shed length if $\phi>\phi_m$ until the concentration is $\phi_m$.

Lattice polygons of length $n$ has linear size $O(n^\nu)$ where $\nu\approx
\sfrac{3}{5}$ is the metric exponent in three dimensions [@Flory69] (a more accurate estimate is $\nu \approx 0.587597(7)$ [@Clisby10]).  Effects of the confining cube will become important when $n^\nu \sim L$. The osmotic pressure should vanish at this point; the result is that $\phi_m \sim L^{1/\nu}/L^3 =
L^{1/\nu-3}$. As $\phi\to 0^+$, $\Pi(\phi;K) \sim L^{-3}$. Using the Flory value for $\nu$ and then plotting $\Pi(\phi;K)\,L^3$ as a function of $\phi\,L^{3-1/\nu} \approx \phi\,L^{4/3}$ should collapse the data in Fig.  \[figure6\].  This is shown in Fig.  \[figure7\], although there are still finite size corrections.  Extrapolating the zeros of the curves in Fig.  \[figure7\] gives $\phi_0 \simeq 0.149\, L^{-4/3}$, $\phi_m \simeq 0.286\, L^{-4/3}$. Since the osmotic pressure vanishes at these concentrations the equilibrium lengths at which the osmotic pressure vanishes are $n_0 \simeq 0.149\, L^{5/3}$ and $n_m \simeq 0.286\, L^{5/3}$. The osmotic pressure of the unknot goes throuh a minimum at $\phi_c \simeq 0.209\, L^{-4/3}$.

The osmotic pressures of compressed lattice knots at low concectration and of knot types $3_1$ and $4_1$ are plotted in Fig.  \[figure8\].  Here the osmotic pressures are monotone increasing with concentration $\phi$, passing through zero at a critical concentration $\phi_0$. Rescaling the data in the same way as in Fig.  \[figure7\] gives Figs.  \[figure9\] and \[figure10\].  This shows that for $3_1$, $\phi_0\, L^{4/3} \simeq 3.94\, L^{-4/3}$, and for $4_1$, $\phi_0\, L^{4/3} \simeq 4.48\, L^{-4/3}$. For $\phi<\phi_0$ the osmotic pressure is negative and the lattice knot will grow to an equilibrium length $n_0 \simeq 3.94\, L^{5/3}$ for $3_1$ and $n_0 \simeq 4.48\, L^{5/3}$ for $4_1$.

Conclusion
==========

In this letter a numerical simulation of compressed lattice knots as a model of an entangled ring polyer show that the osmotic pressure is a function of knot type.  Since the level of entanglements is a function of knot type, these results support the notion that the properties of confined biopolymers, such as DNA, is a function of the level of entanglement if the biopolymer is confined or compressed in a narrow space, or adsorbed on a membrane. 

The numerical data also show that the rescaled osmotic pressure vanishes at a critical concentration, as shown in Figs.  \[figure7\], \[figure9\] and \[figure10\].  The critical concentrations are functions of knot types.  In the case of the unknot there are two critical concentrations where the osmotic pressure vanishes.  At these concentrations the lattice unknot has an equilibrium length, but at the lower critical concentration this is unstable, and the unknot will tend to grow or evaporate.  At the higher critical concentration the equilibrium length is stable.  The situation is not the same for the trefoil and figure eight knot types.  In these cases there is one stable equilibrium at a critical concentration which is dependent on knot type. 

Acknowledgements {#acknowledgements .unnumbered}
================

EJJvR acknowledges support from NSERC(Canada) in the form of Discovery Grant RGPIN-2014-04731. 

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