Modelling a Physical Gel

Gels are polymer networks formed by cross-linking polymeric strands. Cross-linking is generated by, for instance, the presence of cross-linkers, such as proteins in biolog- ical networks [Broedersz and MacKintosh, 2014], or a high concentration of polymer bundles, such as in agarose gels [Pernodet et al., 1997], which is one of the most fa- mous and broadly used examples of gels [Ross, 1964,Stellwagen, 2009,Viovy, 2000]. Agarose gel is commonly used as a buffer to perform gel electrophoresis experiments in order to separate DNA and other bio-polymers. This technique relies on the fact that charged polymeric strands with different lengths, topology or molecular weight possess different mobilities when moving through porous media [Calladine et al., 1991,Calladine et al., 1997,Viovy, 2000]. The main feature that controls how a gel separates molecules with different sizes is the (typical) mesh sizeξg [Calladine et al., 1997]. The more peaked the distribution of pore sizes is, the more uniformly the gel acts as a barrier for molecules larger than the pore size. In practice, agarose gels present a broad distribution of pore sizes, which becomes narrower and centred around smaller pore sizes as the concentration of agarose increases [Maaloum et al., 1998]. In addition, the electrophoretic trapping of circular DNA in agarose gels at strong fields suggests that gels possess dangling fibres which can thread through, and impale, circular DNA and other bio-polymers possessing closed contours [Cole and ˚Akerman, 2003,Stellwagen and Stellwagen, 2009].

The Young’s modulus of agarose is roughly in the regionEagar= 0.1−1 MPa and

depends on the agarose concentration. In any case, it is around 300 times smaller than the one of DNA:EDNA'300 MPa [Marko and Cocco, 2003,Kolahi et al., 2012].

Nonetheless, the persistence length of single agarose fibre of thicknessσagar '2nm

has been found to be lagarp ' 10 nm [Guenet and Rochas, 2006]. Agarose bundles in the sol/gel phase usually form by aggregating a number nf ' 10−20 of single fibres into “fibrils”, which can then display a thicknessσfibril'20−40nm[Pernodet

et al., 1997] and much higher persistence lengths. Even by assuming a simple linear scaling of the persistence length of the bundle as a function of nf, justified by weak interaction among fibres [Mogilner and Rubinstein, 2005], the persistence length of a fibril can still reach valuesl_pbundle'100−200nm, much higher than than of DNA.

3. Molecular Dynamics Models 36

Figure 3.5: Sketch of the gel structure, modelled as(a)perfect rigid cubic lattice and(b)

with dangling ends. Modelling a gel as a rigid mesh is particularly appropriate for artificial gels made of solid nano-wires [Rahong et al., 2014]. Including dangling ends in the model can be of paramount importance when investigating gel electrophoresis of ring polymers (see Ch. 6).

The small Young’s modulus of the whole material is often explained as a conse- quence of the presence of flexible joints in the agarose gel matrix, while the single fibrils and dangling ends [Guenet and Rochas, 2006] are generally considered stiff.

Recently developed artificial gels made of solid nano-wires [Rahong et al., 2014] provide a more selective pore size distribution, and possess a very large Young’s modulus (Ewires '100 GPa) which makes the network less susceptible to deforma-

tions.

From a theoretical point of view, a gel can be thought of as a collection of strands which create a 3D mesh. For simplicity, this mesh is often approximated as a perfect (cubic) lattice [Weber et al., 2006b], with a lattice spacing equal to the average mesh sizehξgi. In this case, the pore size distribution isp(ξg) =δ(ξg− hξgi). In first approximation, the gel structure can also be considered rigid,i.e. completely static, which is a very good approximation in the case of artificially made gels of solid nano-wires.

In Ch. 4, where I will focus on detecting threadings in systems of rings, I will be modelling the gel as a rigid, perfect, cubic mesh (as in Fig. 3.5(a)). In this case, the microscopic structure of the gel is not expected to play a crucial role in the diffusion of rings. In addition, the fact that the gel is perfect eases the formulation of an algorithm that can unambiguously detect the threadings.

In other cases, as in Ch. 6, modelling the gel microscopic structure in detail is important. For instance, the electrophoretic mobility of ring polymers in gel has been found to be strongly dependent on the presence of dangling fibres [Mickel et al., 1977,Turmel et al., 1990,˚Akerman and Cole, 2002,Cole and ˚Akerman, 2003, Stellwagen and Stellwagen, 2009]. Because of this, in Ch. 6, where I will focus on the electrophoretic mobility of knotted ring polymers [Trigueros et al., 2001,Arsuaga et al., 2002], the gel will be modelled as an imperfect mesh, by halving some of the edges forming the cubic lattice with a certain probabilityp (see Fig. 3.5(b)).

3. Molecular Dynamics Models 37

By tuning the probabilitypone can directly regulate the average number of dan- gling ends that can be found in the system, although their length will be fixed to half the lattice spacing, i.e. lde =ξg/2. Unless otherwise stated, the gel lattice spacing

ξg has been chosen to reproduce the pore size of an agarose gel at 5% [Pernodet

et al., 1997] or of an artificial gel after 3 growth cycles [Rahong et al., 2014], i.e.

ξg = 200nm = 80σ in MD units. While this value is unusually high for a standard experiment of gel electrophoresis, it is closer to the typical pore size found in high resolution gel electrophoresis [Stellwagen and Stellwagen, 2009]. The mesh structure is, unless otherwise stated, made up by static beads of sizeσg = 10σ '25nm, which is compatible with the diameter of either agarose bundles (30 nm [Pernodet et al., 1997]) and the nano-wires (20nm [Rahong et al., 2014]).

In general, I will be considering gels whose lattice spacing is comparable with the polymer’s Kuhn length, this is because, from a physical perspective, lattice spacings much greater than the Kuhn length can leave the gel so sparse that the rings rarely encounter it. Alternatively, for lattice spacings much shorter than the Kuhn length, the simulation includes an increasingly large fraction of passive gel monomers, which tend to increase the volume fraction of the system and hence limit the concentration of rings that can be studied efficiently using numerical schemes.

It looks as if it was a bride, walking down the isle, while her dress is being pulled back by flower girls whose dresses are also being pulled by flower girls whose dresses are pulled by other flower girls . . .

A. Y. Grosberg

4

Threading Rings

Contents

4.1 Threading of Rings in a Gel . . . 39

4.1.1 Detecting Threadings between Rings . . . 41