Universidade de Aveiro Departamento de F´ısica, 2010
Pedro Emanuel
Santos Silva
Modela¸
c˜
ao da dinˆ
amica de filamentos el´
asticos
Modelling the dynamics of elastic filaments
Universidade de Aveiro Departamento de F´ısica, 2010
Pedro Emanuel
Santos Silva
Modela¸
c˜
ao da dinˆ
amica de filamentos el´
asticos
Modelling the dynamics of elastic filaments
Disserta¸c˜ao apresentada `a Universidade de Aveiro para cumprimento dos requisitos necess´arios `a obten¸c˜ao do grau de Mestre em F´ısica, realizada sob a orienta¸c˜ao cient´ıfica do Dr. Fern˜ao V´ıstulo de Abreu, Professor Auxiliar do Departamento de F´ısica da Universidade de Aveiro, e a co-orienta¸c˜ao do Dr. Ricardo Sim˜oes, Investigador do Instituto de Pol´ımeros e Comp´ositos da Universidade do Minho e Professor Associado do Instituto Polit´ecnico do C´avado e do Ave.
o j´uri / the jury
presidente / president Armando Neves
Professor Auxiliar da Universidade de Aveiro vogais / examiners committee Fern˜ao V´ıstulo de Abreu
Professor Auxiliar da Universidade de Aveiro Ricardo Sim˜oes
Investigador do Instituto de Pol´ımeros e Comp´ositos da Universidade do Minho e Professor Associado do Instituto Polit´ecnico do C´avado e do Ave Andrea Parisi
agradecimentos / acknowledgements
Quero aproveitar a oportunidade para agradecer `a minha fami´ılia e `a Astrid por todo o apoio que me deram ao longos deste anos.
Agrade¸co tamb´em a todos os meus colegas de curso, em especial ao Daniel e ao Alexandre, que estiveram sempre dispostos a ajudarem-me em qualquer d´uvida relacionada com esta tese.
Quero agradecer ao Professor Ricardo Dias, pelas suas ideias (e tamb´em pela sua paciˆencia) que permitiram-me dar passos importantes no trabalho desenvolvido.
Agrade¸co ao meu orientador, Professor Fern˜ao V´ıstulo de Abreu, pela forma incessante como encaminhou esta tese. Quero tamb´em agrade-cer ao meu co-orientador, Professor Ricardo Sim˜oes, pelas suas sim-ula¸c˜oes que me ajudaram a conceptualizar o algoritmo.
Agrade¸co `a FCT pelo financiamento do projecto FCT PTDC/CTM/101776/2008.
Resumo Nesta tese de mestrado desenvolveu-se um algoritmo para simular o movimento de filamentos el´asticos em trˆes dimens˜oes. O modelo ´e composto por uma cadeia de massas acopladas por for¸cas de elonga¸c˜ao, flex˜ao e tor¸c˜ao. As equa¸c˜oes que descrevem a dinˆamica do sistema foram deduzidas a partir da energia potencial associada aos v´arios graus de liberdade. Nesta tese analisa-se um conjunto de resultados num´ericos obtidos em diversos sistemas, de forma a validar a simula¸c˜ao. Foram obtidas curvas para ensaios de for¸ca-extens˜ao tendo como final-idade a compara¸c˜ao com resultados da F´ısica estat´ıstica de pol´ımeros. Nesta tese h´a tamb´em o cuidado de apresentar as t´ecnicas num´ericas desenvolvidas com vista `a simula¸c˜ao, visualiza¸c˜ao e at´e percep¸c˜ao au-ditiva dos resultados obtidos, com o prop´osito de que este texto possa servir um potencial leitor interessado em iniciar-se nesta ´area.
Esta tese deve tamb´em ser vista como um ponto de partida para a investiga¸c˜ao de novos fen´omenos f´ısicos associados a pseudon´os f´ısicos (Physical PseudoKnots PPKs). Os mecanismos dos pseudon´os f´ısicos procuram explicar, de uma forma inovadora, o aparecimento de cer-tos entanglements localizados que ocorrem na Natureza e que n˜ao requerem explicitamente interac¸c˜oes qu´ımicas.
Abstract In this master’s thesis an algorithm for modelling a three-dimensional elastic filament was developed. The model considers chains of beads that are constrained by stretching, bending and torsional forces. Equa-tions that describe the dynamics of the system were derived from po-tential energies related to the different degrees of freedom. A set of numerical results that were obtained in several systems is analysed in this thesis, as a test bed for the algorithm. Force-extension curves of filaments were obtained to compare the simulated results with the theoretical models from statistical physics of polymers. In this thesis, numerical techniques are also presented in order to perform the simu-lation, visualization and audition of the obtained results, so that this thesis may become an useful guide to a beginner in this field.
This thesis should also be regarded as the starting point for the inves-tigation of new physical phenomena related to physical pseudoknots (PPKs). PPKs mechanisms seek to explain, in an innovative way, the arising of certain localized entanglements that occur in nature and which do not explicitly require chemical interactions.
Contents
Contents i
List of Figures iii
List of Tables v
Introduction 1
1 Modelling a filament 3
1.1 Forces derivation . . . 3
1.1.1 Forces due to stretching . . . 5
1.1.2 Forces due to bending . . . 5
1.1.3 Forces due to torsion . . . 7
1.1.4 Other potentials . . . 8
1.2 Synopsis . . . 9
2 Simulations 13 2.1 Computer and graphical considerations . . . 13
2.1.1 Numerical Integration . . . 13
2.1.2 Computer graphical implementation . . . 14
2.1.3 Sound production . . . 14
2.2 Mass-spring systems . . . 15
2.3 Strings . . . 18
2.3.1 A guitar string . . . 19
2.3.2 A xylophone bar . . . 19
2.3.3 An awkward guitar string . . . 21
2.4 Open rings . . . 22
3 Elasticity models of polymers 25 3.1 The worm-like chain model . . . 25
3.1.1 F → ∞ limit . . . 26
3.2 Force-extension simulations . . . 29
3.2.1 2D-filament force extension (without torsion) . . . 29
Conclusion 33
Perspectives 35
A Torsional term derivation of the force 37
B Solution of one type of nested radicals 39
List of Figures
1.1 Representation of the frames on each bead: ˆxi is the normalized vector that
goes from bead position i to bead i + 1 position; ˆyi ≡ ˆxi−1× ˆxi/kˆxi−1× ˆxik
and ˆzi ≡ ˆxi × ˆyi. . . 4
1.2 Schematic representation of local coordinates associated to stretching (si =
ksik), bending (βi) and torsion (τi). . . 4
1.3 The force resulting from stretching can be seen as a deformation of linear strings. A movement of bead i leads to a deformation of two springs which results in two elastic forces. The numbering on the bottom figure have correspondence with the two contributions in equation (1.7). . . 6 1.4 The force resulting from bending can be seen as a deformation of toroidal
springs. A movement of bead i produces a deformation on three springs producing four forces applied on bead i. The numbers in the figure have correspondence with the four contributions in equation (1.16). . . 7 1.5 Illustration of the torsional force. The torsion along segment si is illustrated
by a spring binding two planes. A displacement of the two planes, and subsequent spring deformation, causes an opposing torsional force. . . 8 2.1 A simple set of instructions (Left ) producing the graphical example on
the Right. This type of graphical tool visualization can be used to ob-tain straightforward representations and animations of filaments in motion. Code in green is optional. . . 14 2.2 System of N bodies of mass m connected by N = N + 1 springs, with
stretching constants k. . . 15 2.3 Modes of transverse oscillation for mass-spring systems with different
num-ber of masses N (dots). The numnum-ber of modes are to the numnum-ber of beads. As more masses and springs we add, the system starts to behave like a continuum string. . . 17 2.4 (a) Some string configurations after pulling bead 50, at the instant t = 0s;
(b) FFT spectrum obtained from the signal produced from the motion of all beads; (c) FFT spectrum of each bead for a guitar string; (d) FFT spectrum of all beads for a string pulled in the bead 10; (e) n-frequency dependency; (f ) Amplitude decay of harmonic frequencies. . . 20
2.5 (a) Fourier spectrum for the signal produced by all beads in a xylophone bar; (b) Inharmonic number versus frequency dependency, exhibiting pro-nounced inharmonicity; (c) Fourier spectrum for the signal produced by all beads in an awkward guitar string; (d) decay of the amplitude decay of the harmonic coefficients exhibiting several decays lifetimes. . . 21 2.6 Three simulations of open rings with external forces applied in the ends. In
the red ring, forces are applied in the tangential direction of the ends. In the green ring, the ends are pulled perpendicular to the ring plane. The yellow ring ends are pulled with a force direction intermediate of the other two. Dashed lines are out of the ring plane. . . 23 2.7 Oscillation of an open ring with opposite (a) parallel forces, (b) with
per-pendicular forces, and (c) a combination of the parallel and perper-pendicular forces acting at the ends. . . 24 3.1 Some chain models used in statistical physics of polymers – (a) gaussian
chain model, GC, (b) freely-jointed chain model, FJC, and (c) worm-like chain model WLC . . . 26 3.2 Force-extension data (red crosses) for λ phage dsDNA pulled by magnetic
beads in 10 mM N a+ buffer. The data are fit to a WLC model solved numerically (WLC exact) or using equation (3.12) (WLC interpolated), as-suming lp = 53 nm. The FJC curve assumes l0 = 2lp = 106 nm. The
Hooke’s law curve from equation (3.11) [20]. . . 27 3.3 Filament produced randomly. . . 30 3.4 Force-extension simulation data (M, ◦, ♦, and O) for a randomly generated
filament with L = 0.15m, d = 1 × 10−4m and different Young’s modulus. The data are fit by (a) the equation (3.10) and (b) by the equation (3.12). 30 3.5 Force-extension responses for an helix with Young’s modulus E = 1GPa and
shear modulus G = 1, 10, and 100GPa . . . 32 3.6 Intertwine of helical structures. . . 36 B.1 (left) Limit results for other combinations of the present nested radical.
In the (right) figure colours are used instead of symbols (−1 = yellow, 0 = black and 1 = green) a fractal behaviour is obtained, Figure B (right). 40
List of Tables
1.1 Schematic representation of the terms of each force. The filled circumfer-ences are the bead i and the colours have direct correspondence with the equations below/above. . . 10 2.1 Comparison of frequencies obtained by an analytical calculation, fteo, and
the computer simulation, fexp, for several spring-mass systems with different
Introduction
There has been an upsurge interest in modelling elastic filaments, due to the potential biological and nanotechnology applications. Proteins [1] and DNA [2] can be seen as elastic filaments, and understanding their elastic properties can be vital to understand how they interact or move. The interest on the elastic properties of filaments has also been growing due to today’s capacity to perform elastic experiments at the nanoscale [3]. With atomic force microscopes (AFM) it is possible to perform force-extension assays [4] measuring forces at the piconewton level and extensions on the nanoscale. AFM could also be used for assembly in what could be can nanoengineering. Understanding the forces involved in these processes, and the dynamics they impose is crucial to accomplish new developments. Curiously, another field that has been devoting a considerable effort in modelling the dynamics of elastic filaments has been in the area of computer graphics animations [5, 6, 7]. Filament dynamics is essential to simulate the real movement of hair, the swinging of trees, etc. Of course, filament dynamics is also studied in many other fields of more traditional sciences, such as chemistry or mechanical and civil engineering. In most cases, the dynamics of elastic filaments is analytically untreatable leading to non-linear equations. For this reason, computational approaches have been extensively developed.
In this thesis I have developed a simple approach to model elastic filaments. All equa-tions were derived starting from the basic classical mechanics equaequa-tions and all important steps are provided, so that it may become an useful guide to a reader with basic knowledge of physics and mathematics.
This thesis is organized as follows:
• In Chapter 1, I will explain the model filament and the mathematical approach to derive the different force equations.
In the first section I present the main assumptions of the model, I will introduce the frame of each bead and then I will discuss the strategy to derive expressions for the forces applied on each point of the filament. Different forces – stretching, bending and torsion – will be considered. Some hints are given to derive other physical interactions, like electrostatic interactions, or the effect of periodic torsional forces. from different nature of the considered ones.
• In Chapter 2, I will present results from simulations testing our approach.
Before starting simulations, in the first section, I discuss some important points concerning Matlab implementations of the algorithm and some aspects of the 3D
representation. In the following section three examples of application are presented. – a mass-spring system: a trivial example from mechanics, for which an analytical
solution can be found;
– a model of strings under an applied tension – this illustrates how the program works in a continuous example where strings, with different physical properties, are tested in the context of the generation of musical sounds;
– systems of open rings – these examples are a starting point for modelling an helix, with bending and torsional movements.
• In Chapter 3, I will explain some polymer models and compare them with this ap-proach.
In the last chapter, I will make a brief introduction of important models to polymer dynamics. Next, the force-extension curve is calculated for the high force limit of a worm-like polymer. Then, two types of filaments are simulated: a random coiled filament confined to a bi-dimensional plane and an helix.
Chapter 1
Modelling a filament
The main goal of this thesis is to develop an adequate algorithm for modelling entangle-ments between filamentary helices. A simple approach is to consider that filaentangle-ments act like classical chains of beads and that their movements are constrained by the material stiff-ness. These considerations, although simple, can be applied in a large number of physical systems, like strings and polymers. These will be presented in Chapters 2 and 3.
The approach adopted here has the intention to be, as Einstein once said, “as simple as possible, but not simpler”. After analysing several options in the literature [7, 8, 9, 10], it was chosen to model a filament as a set of N beads. On each is assigned an index i, and coordinates ri. On each bead it is associated a local frame. The first axis is defined by the
normalized vector joining consecutive beads, ˆxi ≡ (ri+1− ri)/kri+1− rik. The second axis
is directed along the curvature and is given by ˆyi ≡ ˆxi−1× ˆxi/kˆxi−1× ˆxik 6= 0. The third
is orthogonal to former two, ˆzi ≡ ˆxi× ˆyi, see Figure 1.1. Setting local frames in this way
help in the forces derivation. For example, stretching forces results only from variations along the ˆxi axis.
Next, I will show the basic strategy used in deriving the different types of forces – stretching, bending and torsion – on each bead and then I will derive their mathematical expressions. Afterwards, there is a brief section concerning forces related to other types of potentials. A brief summary can be found in the last section.
1.1
Forces derivation
The derivation of dynamical equations requires explicit formulas for the forces acting on each bead. This is done by using the fundamental relation between forces and the potential energy in conservative systems,
Fi = −∇riV, (1.1)
where ri is the position vector of bead i. The force acting on bead i is derived using
equation (1.1) considering the derivative relative to ri. However, the potential energy is
Figure 1.1: Representation of the frames on each bead: ˆxi is the normalized vector that
goes from bead position i to bead i+1 position; ˆyi ≡ ˆxi−1× ˆxi/kˆxi−1× ˆxik and ˆzi ≡ ˆxi× ˆyi.
are used, and consequently one has to apply the chain rule on equation (1.1) relatively to these coordinates. The total force acting on bead i can thus be written as a sum of three main contributions: Fi = − N X j=1 ∂V ∂λstr j ∂λstr j ∂ri + ∂V ∂λbend j ∂λbend j ∂ri + ∂V ∂λtors j ∂λtors j ∂ri ! . (1.2)
Here λj are local coordinates associated to stretching, bending and torsion. The stretching
potential depends on the distance between beads and consequently λstr
j = ksjk ≡ sj.
Similarly, the local bending coordinate is the angle between two consecutive segments, λbend
j = βj and the local torsion coordinate is the angle between consecutive planes (formed
by the consecutive segments), λtors
j = τj. In Figure 1.2 a graphical representation is shown
of these quantities.
In this thesis, I will always consider a linear stress-strain responses [11] of the fila-ment. This corresponds to a general form of Hooke’s law, where harmonic potentials are considered [9]. Even other potential energy dependencies could also be considered, not considering couplings between stretching, bending and torsional modes. For instance, in molecular dynamics, usually a periodic potential is considered to modelling torsional movements [12].
1.1.1
Forces due to stretching
The application of the general method can be made more transparent in the derivation of formulas for the stretching forces. The potential energy due to stretching is:
Vstr = 1 2 N −1 X i=1 ki(si− seqi ) 2 = 1 2 N −1 X i=1 ki(∆si) 2 , (1.3)
where ki = k = ESN/L is the stretching constant, E is the Young’s modulus, S = πR2
is the area of a filament segment with circular cross section of radius R and the contour length L . The segment length si is given by
si = s X ν=x,y,z rν i+1− riν 2 , with rνi ≡ ri· ˆνi (1.4)
Straightforward application of equation (1.2) leads to: Fstri = − ∂V str ∂si−1 ∂si−1 ∂ri +∂V str ∂si ∂si ∂ri = − ki−1∆si−1 ∂si−1 ∂ri + ki∆si ∂si ∂ri . (1.5) Using: ∂si−1 ∂rν i = r ν i − rνi−1 si−1 = s ν i−1 si−1 , ∂si ∂rν i = −r ν i+1− riν si = −s ν i si (1.6) and that, by definition, ˆxi ≡ si/si, we finally obtain
Fi = −ki−1∆si−1ˆxi−1+ ki∆siˆxi = 1stri + 2stri . (1.7)
The physical meaning of this result can be understood with the help of Figure 1.3. When bead i moves to the left, ∆si−1 is negative and consequently the first term corresponds
to a force to the right, along the direction of deformed segment. Simultaneously ∆si
increases and is positive producing another force to the right along the same direction (second contribution).
1.1.2
Forces due to bending
The contributions to the potential energy due to bending are given by
Vbend = 1 2 N −2 X i=1 αbendi (βi − βieq) 2 = 1 2 N −2 X i=2 αbendi (∆βi)2. (1.8) where αbend
i = αbend = EIkN/L is the bending stiffness and Ik = πR4/4 is the second
moment of inertia about a parallel axis. The second term in equation (1.2) then leads to: Fi = − ∂Vbend ∂βi−1 ∂βi−1 ∂ri +∂V bend ∂βi ∂βi ∂ri + ∂V bend ∂βi+1 ∂βi+1 ∂ri = (1.9a) = −
αi−1bend∆βi−1
∂βi−1
∂ri
+ αbendi ∆βi
∂βi
∂ri
+ αbendi+1 ∆βi+1
∂βi+1
∂ri
Figure 1.3: The force resulting from stretching can be seen as a deformation of linear strings. A movement of bead i leads to a deformation of two springs which results in two elastic forces. The numbering on the bottom figure have correspondence with the two contributions in equation (1.7).
The dependency of βi on the coordinates is established through the relation
βi ≡ ∠ (ˆxi−1, ˆxi) = arccos
si−1· si
si−1si
(1.10)
The most difficult part in the derivation consists in calculating and simplifying the three derivatives in equation (1.9b). Here we present the calculations for one of them:
∂βi−1 ∂rν i = − 1 p1 − cos2β i−1 1 si−2si−1 ∂ ∂rν i
(si−2· si−1) + si−2· si−1
∂ ∂rν i 1 si−2si−1 ; (1.11) ∂ ∂rν i (si−2· si−1) = X µ=x,y,z ∂ ∂rν i sµi−2 rµi − ri−1µ = X µ=x,y,z sµi−2δµν = sνi−2; (1.12) ∂ ∂rν i 1 si−2si−1 = 1 si−2 ∂ ∂rν i (si−1) −1 = (si−1) −2 si−2 ∂si−1 ∂rν i = 1 si−2(si−1) 2 sνi−1 si−1 ; (1.13) ∂βi−1 ∂rν i = − 1 sin βi−1 1 si−2si−1 ∂ ∂rν i
(si−2· si−1) + si−2· si−1
∂ ∂rν i 1 si−2si−1 (1.14a) ν ν
Figure 1.4: The force resulting from bending can be seen as a deformation of toroidal springs. A movement of bead i produces a deformation on three springs producing four forces applied on bead i. The numbers in the figure have correspondence with the four contributions in equation (1.16).
Repeating the same process for the other derivatives and substituting in equation (1.9b), we obtain:
Fbendi = αbendi−1∆βi−1
ˆ xi−1× ˆyi−1 si−1 − αbend i ∆βi ˆxi−1 si−1 + ˆxi si
× ˆyi+ αbendi+1∆βi+1
ˆ
xi+1× ˆyi
si
(1.16a)
= 1bendi + 2bendi + 3bendi + 4bendi (1.16b)
A better understanding of the physical meaning of this result can be gained with the help of Figure 1.4. When bead i moves upward, ∆βi±1are positive and ∆βi is negative. Vectors
ˆ
yi±1 and ˆyi are perpendicular to the plane of motion and point upward and downward,
respectively. The resulting forces gives contributions perpendicular to the segment ˆxi−1
(contributions 1 and 2) and to the segment ˆxi (contributions 3 and 4), to restore the
original configuration.
1.1.3
Forces due to torsion
Like in the previous subsections, an harmonic potential associated to the torsional elastic modes is considered:
Vtors= 1 2 N −2 X i=2 Ci(τi− τ eq i ) 2 = 1 2 N −2 X i=2 Ci(∆τi) 2 , (1.17)
where Ci = C = GI⊥N/L is the torsional stiffness, G is the shear modulus and I⊥ = πR4/2
is the second moment of inertia about the perpendicular axis. The torsional component appears with the variation of the angle between the planes defined by the segments {si−1, si}
and {si, si+1}: τi ≡ ∠ (ˆyi, ˆyi+1) = arccos (si−1× si) · (si× si+1) k (si−1× si) kk (si× si+1) k (1.18)
Figure 1.5: Illustration of the torsional force. The torsion along segment si is illustrated
by a spring binding two planes. A displacement of the two planes, and subsequent spring deformation, causes an opposing torsional force.
Figure 1.5 shows two frames of a filament with a different torsional angle. Once again, the next step is to calculate all the derivatives in the contributions:
Fi = − ∂Vtors ∂τi−2 ∂τi−2 ∂ri + ∂V tors ∂τi−1 ∂τi−1 ∂ri + ∂V tors ∂τi ∂τi ∂ri + ∂V tors ∂τi+1 ∂τi+1 ∂ri = (1.19a) = − Ci−2∆τi−2 ∂τi−2 ∂ri + Ci−1∆τi−1 ∂τi−1 ∂ri + Ci∆τi ∂τi ∂ri + Ci+1∆τi+1 ∂τi+1 ∂ri (1.19b)
At this point, it is convenient to mention that, although the mathematics involved is relatively simple, considerable calculations are involved. The calculation for one of the derivatives can be found in the Appendix A. Deriving all the four contributions the following expression for the torsional force obtained:
Ftorsi = Ci−2∆τi−2
ˆti−2× ˆyi−1 pi−1 × ˆxi−2 − Ci−1∆τi−1 ˆ ti−1× ˆyi−1 pi−1 × ˆxi−2+ ˆti−1× ˆyi pi × (ˆxi−1+ ˆxi) ˆ t × ˆy ˆt × ˆy
collisions. Another effect that can be of interest is related to modelling torsional forces for large deformations. This will be discussed in the end of this section.
The electrostatic potential is given by:
Vel= 1 2 N −1 X i=1 Kielqiqi+1 si , (1.21) where Kel
i is the Coulomb constant. Applying the chain rule as we did before, in equation
(1.2), we rewrite: Feli = − ∂V el ∂si−1 ∂si−1 ∂ri +∂V el ∂si ∂si ∂ri . (1.22)
The first part of the derivative gives ∂Vel ∂si = −Kielqiqi+1 s2 i (1.23)
and second part is the same result as obtained before for the stretching force, equation (1.6).
For large torsional deformations, it is important to highlight that different models could be considered. Above we considered a simple harmonic potential associated to torsional modes. This can be inadequate to describe torsion degrees of freedom in molecules, for instance. In that case the elastic potential should be periodic instead of growing steadily. However this kind of modification can be easily incorporated in our approach. Considering the most commonly used molecular dynamics periodic potential for torsion degrees of freedom: VM D = N −2 X i=2 Kitors(1 − aicos ∆τi) , (1.24)
where here Ktors
i is a molecular torsion constant and ai is the number of bonds connected
to bead i. The force derivation proceeds as in the section 1.1.3. The main difference is now the derivative of the potential becomes:
∂VM D
∂si
= aiKitorssin ai∆τi. (1.25)
1.2
Synopsis
In this chapter I have presented a simple model to compute the dynamical behaviour of an elastic filament. We have introduced a frame of reference attached to each bead, on which:
• ˆxi is a unit vector joining consecutive beads, ˆxi ≡ si/si, where si ≡ ri+1− ri and
• ˆyi is a unit vector perpendicular to the plane formed by consecutive monomers,
si−1× si/ksi−1× sik;
• ˆzi is the unit vector perpendicular to the other two, ˆzi ≡ ˆxi× ˆyi;
• the bending angle is defined by βi ≡ ∠ (ˆxi−1, ˆxi) and the torsional angle by τi ≡
∠ (ˆyi, ˆyi+1).
After the derivation of the different forces – stretching, bending and torsion – it can be concluded that: two, is the minimum number of beads required to model a filament and this filament will only perform stretching movements; bending deformations, require the introduction of a third bead to deform the angle between two consecutive segments; the torsional deformation requires a minimum of four beads, and arises when the plane formed by si−1 and si changes relatively to the consecutive plane formed by si and si+1.
Stretching
Bending
Torsion
Table 1.1: Schematic representation of the terms of each force. The filled circumferences are the bead i and the colours have direct correspondence with the equations below/above.
Fstri =−ki−1∆si−1xˆi−1+ki∆sixˆi Fbendi =αbendi−1∆βi−1
ˆ xi−1× ˆyi−1 s −α bend i ∆βi ˆxi−1 s + ˆ xi s
× ˆyi+αbendi+1∆βi+1 ˆ
xi+1× ˆyi s
In the Table 1.1 a diagrammatic representation of the different terms contributing to the dynamical forces in equation (1.2) is presented. The filled bead represents the bead receiving contributions to the total force, from the other neighbouring beads, when the deformation indicated by the arrow is performed. The deformation is due to a movement of the filled bead. For each contribution this deformation changes only the variable con-tributing to the force (distance between beads, for the stretching, angle between monomers for the bending force and angle between consecutive planes for the torsion contribution). For instance, concerning the stretching force, two contributions should arise, one involving the bead i + 1 and the other the bead i − 1. A less trivial example concerns computing all contributions due to bending. In this case, bead i receives contributions from different deformed angles, involving different beads (Table 1.1 second row). The first contribution involves beads i − 2 and i − 1 and is due to a change in the bending angle αi−1 due to a
movement of the segment si−1 relatively to the segment si−2. The second diagram takes
into account contributions involving beads i − 1 and i + 1. In this case the movement of bead i changes the bending angle αi through changes on two segments, si−1 and si. As
si−1 and si both depends on ri there are two terms associated to this diagram. Finally,
the third diagram leads to the same reasoning as done for the first diagram. A similar reasoning can also be done consider contributions arising from torsion. Below I present the full set of dynamical equations, written with colourings in accordance to those used in Table 1.1 for a comparison. This diagrammatic representation was extremely useful to confirm that no contributions were missing.
Chapter 2
Simulations
After deriving expressions for the forces acting on each bead, it is now necessary to describe the dynamics of the system along the time. In this chapter, simulations that will enable three dimensional visualisation of the dynamics of an elastic filament are developed. In some cases, an algorithm to perceive the sound generated by the filament is also provided. This will add an extra dimension in the simulations and also another way of testing the validity of the simulations. First,the algorithm will be tested with a discrete problem of mass-spring systems. Only stretching forces will be considered. In the case of strings, the same problem will be analysed but tending to the continuum limit and also considering a bending stiffness. The last problem is concerned with the movement of open rings and puts in evidence the effect of bending and torsion.
2.1
Computer and graphical considerations
2.1.1
Numerical Integration
Using the force expressions, in equations (1.7), (1.16a) and (1.20), it is straightforward implement to algorithms for the numerical integration of the equations of motion. There are several numerical methods to perform this integration. For simplicity, we used the Euler-Cromer method that is known to produce good results in conservative systems [13], like a pendulum. The method consists in a modification of the Euler formula, where for updating the positions at time t + δt, the estimate of the velocities at time t + δt is used instead of at time t: vi(t + δt) = vi(t) + Fi(t) mi − γivi(t) δt (2.1) xi(t + δt) = xi(t) + vi(t + δt)δt (2.2)
In equation (2.1) we have also included a contribution, γi, from viscous damping on
2.1.2
Computer graphical implementation
There are several ways to visualize the computed data. I developed methods in Math-ematica to produce quality visualization requiring a simple manipulation. An example with a set of instructions as well as the graphical output is shown in Figure 2.1. Some (minor) modifications would be required to upload a data file containing information of beads positions along the time.
Fig = Animate[ Graphics3D[{
Lighting → “Neutral”, Specularity[1,5], Black, Sphere[{{1, 0, 0}, {2, t, 0}, {3, 0, 0}}, .2], Opacity[.5], JoinForm[“Miter”],
Tube[{{1, 0, 0}, {2, t, 0}, {3, 0, 0}}, .2] , Boxed → False, ViewPoint → Top] ,{t, 0, 1, .5}]
Figure 2.1: A simple set of instructions (Left ) producing the graphical example on the Right. This type of graphical tool visualization can be used to obtain straightforward representations and animations of filaments in motion. Code in green is optional.
2.1.3
Sound production
I also developed numerical methods to convert strings movements into the sounds they can produce. The sound produced by the motion of a string, results from air disturbances produced by all beads. A bead in a string with tension, placed in a xy − plane, will nearly oscillate only in the z − axis projection of the position. In Matlab it is easy to perceive the sound that strings would generate, by using the sound instruction:
m m m
Figure 2.2: System of N bodies of mass m connected by N = N +1 springs, with stretching constants k.
2.2
Mass-spring systems
A well known problem in mechanics is the calculation of the normal modes of mass-spring systems. In order to validate the approach concerning the equations for stretching I simulated several systems and compared the respective FFT spectrum with the analytical solutions.
Let us consider N masses m connected by N = N + 1 springs, having equal stretching constants k, Figure 2.2. Springs are massless and at rest they are equally distributed along a length L = N l0, with an exerted tension T0 = k(seq− l0). These systems of masses and
springs are placed on a xy − plane and oscillate along the z − axis.
In order to find the traverse modes of oscillation of the masses the approximation of small oscillations is usually assumed [14]. The amplitude of oscillation A is smaller than the systems length, A L. The equations of motions are:
m1z¨1+ (2T0/l0) z1− (T0/l0) z2 = 0 m2z¨2+ (2T0/l0) z2− (T0/l0) z3− (T0/l0) z1 = 0 .. . mjz¨j+ (2T0/l0) zj− (T0/l0) zj+1− (T0/l0) zj−1= 0, with j = 2, . . . , N − 1 .. . mNz¨N + (2T0/l0) zN − (T0/l0) zN −1 = 0 (2.3)
where for notation convenience, zi is the third component of the vector si, zi = szi.
Sup-posing that we have a periodical system then, looking at solutions of the kind zi = sin ωit
we obtain an equation that can be solved by the determinant: 2A − B −A 0 · · · 0 0 −A 2A − B −A · · · 0 0 0 −A 2A − B · · · 0 0 .. . ... ... . .. ... ... 0 0 0 · · · 2A − B −A 0 0 0 · · · −A 2A − B = 0, (2.4)
with A = T0/l0 and B = mω2. Several systems were analytically solved to compare their
solutions with my simulations. Frequencies for the analytical and numerical simulations are displayed in Table 2.1. The analytical solutions for the frequencies are of the form of
fteo= ω 2π = 1 2π r χ T ml0 , (2.5)
N fteo= 2π1 q χmlT 0 (Hz) fexp (Hz) χ Analytical Simulation 1 (2) (88.28) (88.3) 2 (1) ; (3) (88.28) ; (152.91) (88.3) ; (152.9) 3 (2) ; 2 ±√2 (161.18) ; (210.59, 87.23) (161.2) ; (210.6, 87.2) 4 3± √ 5 2 ;5± √ 5 2 (225.86, 86.27) ; (265.51, 164.10) (225.9, 86.3) ; (265.6, 164.1) 5 (1) ; (2) ; (3) ; (165.16) ; (233.58) ; (286.07) ; (165.2) ; (233.6) ; (286.1) ; 2 ±√3 (85.49, 319.07) (85.5, 319.1) 7 (2) ; 2 ±√2 ; (305.82) ; (399.58, 165.51) ; (305.9) ; (399.7, 165.5) ; 2 ±p2 ±√2 (424.19, 359.61, 84.38, 240.28) (424.3, 359.7, 84.4, 240.3) (2) ; (378.01) ; (378.1) ; 9 3±√5 2 ; 5±√5 2 ; (432.49, 165.19) ; (508.42, 314.22) ; (432.6, 165.2) ; (508.5, 314.3) ; 2 ± q 5±√5 2 (528.02, 476.32, 83.63, 242.61) (528.1, 476.4, 83.6, 242.7) Table 2.1: Comparison of frequencies obtained by an analytical calculation, fteo, and the
computer simulation, fexp, for several spring-mass systems with different number of masses
N .
where χ is a root for the characteristic polynomial in equation 2.4.
Analytical and simulation results were very close. The worst result had a difference of only 0.1%. This was an important confirmation that our simulations are producing the correct results, at least as stretching interactions are concerned.
An illustration of different transverse modes is shown in the Figure 2.3. From these diagrams it is clear that wave functions for low frequency modes are well approximated by systems with a small number of beads. To simulate high frequency modes more and more beads are required. This is important to retain if one wants to compare numerical
Figure 2.3: Modes of transverse oscillation for mass-spring systems with different number of masses N (dots). The number of modes are to the number of beads. As more masses and springs we add, the system starts to behave like a continuum string.
χ in the equation (2.5, it comes:
f1 = 1 2π r T M L √ 22ϑx. (2.7)
In the continuum approximation we can use the limit (see Appendix B):
lim ϑ→∞2 ϑ v u u t2 − s 2 + r 2 + ... q 2 +√2 = π. (2.8) Resulting in: f1 = 1 2 r T M L. (2.9)
This result can also be achieved by the relationship between the solution of the wave equation, v =pT L/M, and fundamental vibrational mode for a stretched string, L = λ/2 [15].
2.3
Strings
The sound waves generated by musical instruments, like the strings of a guitar or a piano, are a continuum example of filaments, from the everyday life. These examples have been extensively studied, due to their wide interest and complexities [16]. As a result, studying the dynamical effects arising in musical strings can work as a test bed for our numerical simulations.
Undergraduate courses in physics teach that strings oscillate at harmonic frequencies, fn, multiples from a fundamental frequency, f1. However, real strings and other types of
waves, generated by musical instruments, oscillate in a sequence of frequencies that do not strictly follow this simple relationship. This can be seen using the differential wave equation for a string with bending stiffness [17]:
T∂ 2y ∂x2 − EI ∂4y ∂x4 = µ ∂2y ∂t2, (2.10)
where T = σS is the tension on the string, σ is the tensile stress and µ is the linear density. Inserting in equation (2.10) a general solution of the form yn = e−ifntYn, [18], it can be
verified that normal frequencies must follow the relation:
fn = nf1
r
1 + π2n2 EI
T L2, (2.11)
From equation (2.11) it can be concluded that bending stiffness introduces an inhar-monicity factor correction, which increases with the decreasing of the filament length L, the decreasing string tension T and increases with the harmonic number n. It also increases with the filament cross section through I. Inharmonicity produces non-trivial effects on sound perception. This is part of the reason why different instruments sound so different. The effect of bending is more pronounced in instruments like idiophones (for instance, xy-lophones, marimbas) rather than in strings instruments, like acoustic guitars. Of course, in musical instruments sound also depends very importantly on the acoustics of the body of the instrument. This will not be discussed here.
2.3.1
A guitar string
To simulate the sound produced by a guitar string we considered a filament with N = 200 beads, radius r = 0, 25mm, length L = 710mm, Young’s modulus E = 190GPa and linear density µ = 1.55 × 10−3Kg m−1. In Figure 2.4(a), string configurations are shown for an example where a bead, placed at L/4, was pulled at a constant force F = 12N (F = Fexternal − Fstring reaction) applied for 1/300 seconds, and then released. We used a
dissipation coefficient γ = 3s−1 for all beads. The spectrum is shown in Figure 2.4(b). This is the typical representation of a spectrum where the several spectral lines are given by the modulus of the coefficients of the Fourier modes present in the signal. An alternative representation is shown in Figure 2.4(c), where the string configuration for each bead is shown and represented according to the amplitude given by the Fourier coefficients for each mode. This representation clearly shows that the fundamental is the easiest mode to excite, which agrees with the fact that it requires the lowest energy. However, other modes are also present. By plucking the string in a different location, the relative composition of the several modes can be changed, but the frequencies remain the same, as expected, Figure 2.4(d). In terms of sound perception, differences between the different sounds hence generated are noticeable. Sounds are richer when more harmonics are excited. In Figure 2.4(e) we show how the several spectral peaks decay. When all beads are identical, all modes decay with identical time constants. Finally, in Figure 2.4(f ) we display how frequencies vary with the harmonic number. This dependency is linear, which was expected because as mentioned, in the previous section, the bending contribution is not significant in this case. Indeed, for n = 20, deviations from the linear curve represent only 0.2%.
2.3.2
A xylophone bar
The impact of bending can be appreciated by analysing the sound produced by a thick wire. We simulated a filament with N = 200 beads, radius r = 3mm, length L = 500mm, Young’s modulus E = 190GPa and linear density µ = 1.55 × 10−3Kg/m, corresponding to a metallic wire. As can be seen in Figure 2.5(b) frequencies are no longer multiples of the fundamental. For n = 3, there is already a 10% deviation from the harmonic frequency. The third harmonic is easily generated in most strokes and consequently inharmonicity ef-fects should be clearly audible. Indeed, in our simulations, sounds produced with stronger inharmonicity are perceived differently. Inharmonicity perception is a phenomenon arising from complex brain processing, somehow comparable to the effects caused by ambiguous figures in visual perception. Our brain tends to perceive the fundamental frequency in sounds. It does so by performing a kind of spectral analysis of sound, in which the fun-damental frequency is identified, either by its audition or by perception of higher multiple frequencies (also known as overtones). It is possible to produce the sensation that the fun-damental frequency is present, even if it is in fact absent but overtones are not. Ambiguity appears when the ability to extract the fundamental frequency from higher multiples is not present, as happens when inharmonicity is present. In that case, the subject has difficulty in identifying the fundamental frequency, and may perceive a different, pseudo-frequency
(a) (b)
(c) (d)
to the same musical note. However, the sound with more harmonics is not as pure.
(a) (b)
(c) (d)
Figure 2.5: (a) Fourier spectrum for the signal produced by all beads in a xylophone bar; (b) Inharmonic number versus frequency dependency, exhibiting pronounced inharmonic-ity; (c) Fourier spectrum for the signal produced by all beads in an awkward guitar string; (d) decay of the amplitude decay of the harmonic coefficients exhibiting several decays lifetimes.
2.3.3
An awkward guitar string
Our numerical simulations are very flexible in what concerns testing more elaborate systems, like non-uniform strings. We studied the effect of introducing two beads in differ-ent positions and with differdiffer-ent friction coefficidiffer-ents, in the guitar string discussed in section 2.3.1. The impact of the beads in the generated sound depends crucially on where they are placed. If very heavy identical beads are placed at the nodal points of the uniform string for some harmonic n, then beads will tend to hold the string at these nodal positions suppressing frequencies below fn. In a certain sense beads work as filters eliminating a
set of frequencies from the spectrum. If beads are not as heavy, then their movement will couple to the movement of the string and the spectrum becomes more complex, with more
spectral peaks appearing. We simulated a string with one bead, forty times heavier than the remaining, placed at middle. The corresponding spectrum had the same spectral lines for even harmonics as in the previous guitar string. However, new spectral lines appeared because the movement was not totally constrained.
A more awkward sound can be produced if beads are placed in positions that do not cor-respond to nodal positions of the uniform string. When beads have a large mass then they will tend to segment the movement of the string in the composition of the movement due to smaller segments. We placed identical beads at x = 2L/3 and x = 2L/5. The spectrum is shown in Figure 2.5(c). We clearly observe the appearance of many more peaks. The per-ceived sound, in this case, has clearly two different pitches, one lower, corresponding to the first two peaks, and one higher corresponding to the remaining ones. Another interesting phenomenon arises when the modified beads have higher damping constants. Then several modes have different damping rates. As beads have higher damping coefficients, modes more closely related to their movement decay faster. These could correspond to some of the lower frequency modes. As we can see in Figure 2.5(d) the lower pitch modes decay faster, despite having an initial higher amplitude. This results in an earlier suppression of that pitch before other modes. The sound perceived is much richer not only in terms of the number of the spectral peaks present, but also on how they evolve in time.
2.4
Open rings
The previous two example were conceptually bi-dimensional problems. As a first step to study helical filaments, which requires a three-dimensional approach, I will start by addressing the dynamics of simple open rings. The helix can be seen has a set of connected rings without geometrical torsion. The simulations will consist in the application of forces, with specific directions, in the ends of the rings. This section has also the purpose to view how bending and torsional modes affect the movement of an helix when a ring is stretched. The simulated open rings have stiff bond connections, this means that the movement will not depend significantly of stretching.
Considering a system that consists on a chain of N = 13 beads equally spaced on a circumference, as shown in Figure 2.6. The filament is made of steel (E = 200GPa, G = 80GPa and µ = 1.55 × 10−3Kg m−1), have a radius of r = 1mm and has a bead separation
Results from the three numerical experiments with φ = 0, ±π/4 and ±π/2 are pre-sented in Figure 2.7, where φ is the angle between the applied force and the ending segments vectors. These experiments illustrate the impact of the different forces for the movement. In the first case only bending is present, while in the second case only torsion contributes. When φ = ±π/4 both torsion and bending contribute, although, in this case, the ampli-tude of the torsional energy is greater than bending one. Bending and torsion modes are associated to different planes. This will have a crucial implication when we stretch an helix, because different stiffness of the filaments might means completely different behaviours of their dynamics.
For this filament, the segment length have not relatively changed, so it wouldn’t be expected a significant stretching force. In fact, when we stretch a Hookean spring, there is almost no stretching of the filament. There is a competition between bending and torsional movements. This means the string will twist more or less depending on the filament stiffness.
Figure 2.6: Three simulations of open rings with external forces applied in the ends. In the red ring, forces are applied in the tangential direction of the ends. In the green ring, the ends are pulled perpendicular to the ring plane. The yellow ring ends are pulled with a force direction intermediate of the other two. Dashed lines are out of the ring plane.
(a) (b) 0 1 2 3 4 5 x 10−3 0 1 2 3 4 5 x 10 −3 Time (s) Energy (J) E total E stretch E bend E tors 0 1 2 3 4 5 x 10−3 0 2 4 6 8 x 10 −3 Time (s) Energy (J) E total E stretch E bend E tors (c) 0 1 2 3 4 5 x 10−3 0 2 4 6 8 x 10 −3 Time (s) Energy (J) E total E stretch E bend E tors
Chapter 3
Elasticity models of polymers
In the last chapter numerical results were compared with analytical calculations for sev-eral systems. These examples allowed us to confirm that simulations seem to be producing the correct results at least in what bending and stretching contributions are concerned. Our incursion into the field of musical sound production also it has allowed to recapture some complex phenomena like the inharmonicity effects that appear in the xylophone bar. Our field of interest is, in fact, less prosaic (mundane). Our goal is to apply these simula-tions to study the behaviour of polymers, with potential applicasimula-tions in nanothecnology. The recent upsurge of nanotechnology has been one of the reasons why the simulation of single filament dynamics gained increasing attention in the literature. Another source of motivation has been the study of the DNA molecule [20]. From a polymer science point of view, this molecule is very interesting because it is one of the most highly charged polymers known and the double-stranded DNA (dsDNA) has an unusual stiffness, when compared to the single-stranded form (ssDNA) [21]. Furthermore nanotechnology now made avail-able a wide range of techniques, like hydrodynamic drag [22], magnetic beads [23], glass needles [24], optical traps [25, 26] and AFM [27]. Consequently experimental works can now provide important and accurate information to validate theoretical models. For in-stance, force-extension experiments measure the end-to-end distance of a chain subject to an applied force, just as required by the simplest theoretical models. Also, these theoretical models and experiments allows us to test our algorithm in medium to large deformations regimes of a chain, which has not the case of the previous theoretical results, obtained using small deformations approximations.
In this it will be chapter summarized what is known concerning worm-like chain models, and then we analyse how our simulations agree (or not) we these models.
3.1
The worm-like chain model
Considering a polymer made of a chain with N beads, the simplest model to describe these polymer conformations is the freely-jointed chain model (FJC). This model considers a constant distance between consecutive beads equal to l0. This movement has a
correspon-(a) (b) (c)
Figure 3.1: Some chain models used in statistical physics of polymers – (a) gaussian chain model, GC, (b) freely-jointed chain model, FJC, and (c) worm-like chain model WLC
dence to a random walk chain that tends to align with an external applied force. A similar model, the Gaussian chain model (GCM) [28], considers the l0 distance as drawn from a
Gaussian distribution, which corresponds to a chain of Hookean springs. This model has the non-physical property that the filament can be indefinitely pulled. In these two models, the variation of the angles, between the bonds segments, does not change the energy of the chain. Likewise, the polymer can freely rotate.
The worm-like chain model (WLC), also known as the Kratky-Porod chain, takes in account an additional intrinsic stiffness of the nodes. The WLC model describes reasonably a polymer with an helical structure, as the case of DNA [21]. These are phantom chains models, which means they can intersect themselves freely without interacting.
Several force-extension experiments have provided information to test the more fre-quently used models, FJC and WLC. Figure 3.2 shows the characteristic force-extension behaviour of each approximation and experimental results. All the models, in the limit of low forces, describe the polymer has a Hookean spring. In the semi-flexible and rigid-rod limits, the FJC model fails considerably. By contrast, the WLC model provides an excel-lent description of the polymer elasticity in these two limits. Above 10 pN, the end-to-end distances of the experimental results are slightly larger than the WLC model, probably due to a small elongation of the bonds junctions.
Figure 3.2: Force-extension data (red crosses) for λ phage dsDNA pulled by magnetic beads in 10 mM N a+ buffer. The data are fit to a WLC model solved numerically (WLC exact) or using equation (3.12) (WLC interpolated), assuming lp = 53 nm. The FJC curve
assumes l0 = 2lp = 106 nm. The Hooke’s law curve from equation (3.11) [20].
• flexible, L lp,
• semi-flexible, L ∼ lp,
• and a rigid-rod, L lp.
Curving a polymer, with a considerably bending stiffness, has an energy cost, then it tends to adopt a more straight conformation. A flexible polymer, with a low bending rigidity, tends to have a more compact random-coil conformation. In the rigid-rod limit, persistence length and bending stiffness can be related.
The worm-like chain model describes a chain of beads with bending stiffness, where the segments lengths are constant. This means that we need energy to bend two adjacent bonds. The Hamiltonian of this system is given by [30]:
H = −α bend 2 N −1 X i=1 ˆ xi· ˆxi+1. (3.2)
This is similar to the one-dimensional Heisenberg model for the ferromagnets. Taking the assumption that ksjk = l0, taking the continuum limit, N → ∞, l0 → 0 and αbend → ∞
and using −ˆxi−1· ˆxi = (ˆxi−1− ˆxi)2/2 − 1 we get:
H = α bendl 0 2 Z L 0 ∂ˆx ∂s 2 ds = α bendl 0 2 Z L 0 c2(s)d, (3.3)
where c(s) is the curvature of the chain contour and s is the continuum variable that replaces the index i. Considering an external force, Fext= {0, 0, F }:
H = α bendl 0 2 Z L 0 ∂ˆx ∂s 2 ds − Z L 0 Fext· ˆx(s)ds (3.4)
In the approximation of high-force, the tangent of the polymer is nearly pointing in the di-rection of the applied force. Writing the ˆx vector in terms of his parallel and perpendicular components, ˆ x ≈ ˆx⊥ˆx⊥+ 1 − 1 2xˆ 2 ⊥ ˆ xk, (3.5)
and substituting this result in the equation 3.4, we obtain the following harmonic Hamil-tonian: H ≈ α bendl 0 2 Z L 0 ∂ˆx⊥ ∂s 2 + F ˆx2⊥ ! ds − F L. (3.6)
This integral is a standard exercise of path integration. However, the calculation can be made more clear rewriting the Hamiltonian in term of Fourier modes by applying the transformation ˆx⊥ = 2Pqaqcos qs to equation (3.6),
H = LX
q
αbendq2+ F |aq|2− F L (3.7)
Assuming that the polymer is in equilibrium at temperature T , by the theorem of equipar-tition, we get: 2L k 2q 2+ F |aq|2 = d kBT 2 , (3.8)
where d is the dimensionality.
To obtain the the mean end-to-end distance distribution, we have to integrate all the perpendicular contributions from the equation (3.5):
h∆Ri = Z L 0 1 − 1 2|ˆx⊥| 2 ds = L −X q |aq| 2 (3.9a) = L − L Z ∞ k BT dq (3.9b)
From equation (3.10) it can concluded that the force increases with the temperature and decreases with the bending stiffness. This equation is valid in the large force regime. In the limit of small extensions h∆Ri/L 1 the polymer acts as a Hookean spring and its extension is proportional to the applied force:
F = 3kBT 2lp
h∆Ri
L . (3.11)
Marko and Siggia obtained an interpolation and exact solutions to the WLC model. However, the interpolated formulae is most frequently used to fit experimental data because the exact solution has to be determined numerically, minimizing a variational parameter for any given force. The interpolated formula is simply given by:
F = kBT lp 1 4 (1 − h∆Ri /L)2 − 1 4+ h∆Ri L . (3.12)
This interpolation formula provides an acceptable fit for some results. However, the inter-mediate regime of the force has up to 6% of error relatively to the exact solution.
3.2
Force-extension simulations
Force-extension experiments have an important role in the characterization of macro-molecules and polymers. In this section, I will present result of simulations for force-extension experiments. First we will consider filaments with no torsional stiffness, for comparison with the theoretical prediction in the high limit force of the WLC model. Af-terwards we will consider the simulation of helical filaments. In this case torsional stiffness is included and comparison with the previous results will the provided.
3.2.1
2D-filament force extension (without torsion)
The relationship between force and extension will be first tested with a bi-dimensional filament. I started by creating a randomly generated filament with N = 30 segments in a two dimensional plane (Figure 3.4). One end was kept fix and on the other an external force Fext is applied in the same plane of the chain. In this case, beads will always be
on the same plane as no torsional forces are considered. For several applied forces the end-to-end distance when the filament reaches equilibrium was measured.
Figure 3.4 presents typical results of the characteristic force-extension curves, obtained when the Young’s modulus was varied for E = 1, 10, 100, 200 and 1000GPa. Both stretch-ing and bendstretch-ing constants depend on the Young’s modulus, E, accordstretch-ing to equations (1.3,1.8). When E is low, bending rigidity is lower and a bigger extension will be obtained for smaller forces. Besides, as the force is greater the distance between beads increases. For E = 1GPa there is significant extension of the segment’s length, starting at forces ∼ 10−1N. When the chain has higher stiffness, the movement will be more constrained by
Figure 3.3: Filament produced randomly. (a) (b) 0 0.2 0.4 0.6 0.8 1 1.2 10−4 10−3 10−2 10−1 100 x/L F ext (N) E = 1 GPa E = 10GPa E =100GPa E =200GPa E=1000GPa 0 0.2 0.4 0.6 0.8 1 1.2 10−4 10−3 10−2 10−1 100 x/L F ext (N) E = 1 GPa E = 10GPa E =100GPa E =200GPa E=1000GPa
Figure 3.4: Force-extension simulation data (M, ◦, ♦, and O) for a randomly generated filament with L = 0.15m, d = 1 × 10−4m and different Young’s modulus. The data are fit by (a) the equation (3.10) and (b) by the equation (3.12).
Figures 3.4(a) and (b) also present fits to the numerical results by WLC solutions, in the high-force limit, equation (3.10), and the interpolated formula of Marko and Siggia,
In these simulations, stretching stiffness is much larger than bending stiffness. Conse-quently, the chain does not extend significantly beyond x = L. Only, when E = 1GPa can we observe the chain entering an overstretching regime ( x > L). For larger values of E, the force-extension curve seems to diverge at x = L. This apparent divergence is approached quickly for smaller values of the Young modulus, while for larger values filaments stay in an more intermediate regime, whereby segments remain (somewhat) unaligned.
By comparing the numerical results with the theoretical WLC solutions it is clear to conclude that they differ considerably in the intermediate to high force regimes. This can be due to several reasons that require further investigations. Unfortunately, I was not able to answer these issues in time. Consequently only a set of hypothesis and suggestions are provided here. First I could point that this difference may be due to a finite system effects. Larger systems would have to be simulated and a trend extracted to extrapolate continuous limit results. Another cause for the difference could be due to the fact that in theoretical models stretching is assumed infinite. I.e., no deformations occur along the segments. In my simulations stretching is always present, even if I had set it to high values. As a result elongations should be expected to be always larger than expected. Probably the most reasonable explanation could be that in our simulations we are not considering stochastic forces from the surrounding medium. Entropic forces would necessarily diffi-cult the alignment along the applied force direction. Consequently the force required to produce a given extension should be larger. In principle it is easy to test this hypothesis, since introducing a random stochastic force is straightforward in the algorithm. In case this explanation is the valid one, then it would mean that the simulations presented here described systems for which thermal effects were negligible, i.e., kBT << Ebend<< Estret,
while those described by WLC models describe systems for which Ebend<< kBT << Estret
3.2.2
Stretching the helix
This thesis was motivated by the study of entanglements in helices. Helices are nec-essarily 3D structures, and consequently we may wonder whether torsional effects may play a role and induce different behaviour. Although I did not have enough time to de-velop thorough studies on this topic, I will show preliminary studies involving helices. In this subsection I show force-extension curves obtained with an helical filament. The main purpose of this study was to see how torsion affects the elongation of an helix.
I generated a helix with two loops and N = 55 segments. The initial end-to-end distance was equal to L0 = 0.1m and the cross-section was d = 1 × 10−3m. As in the
previous example, one of the ends was kept fix and on the other an external force Fext
was applied all the time along the axis of the helix. Simulations considered helices with different shear modulus, to study how this parameter affected the force-extension response of the helix.
The main result from these experiments can be understood from an analysis of Figure 3.5. This figure shows force-extension responses for an helix with Young’s modulus E = 1GPa, but different shear modulus. Since the several curves do overlap we conclude that torsion deformations do not affect the response. This conclusion is maintained for filaments
0.8 1 1.2 1.4 1.6 10−2 10−1 100 101 102 103 x/L F ext (N) G = 1 GPaG = 10GPa G =100GPa
Figure 3.5: Force-extension responses for an helix with Young’s modulus E = 1GPa and shear modulus G = 1, 10, and 100GPa
with other Young’s modulus. This result is interesting although it would be important to develop further simulations to ascertain its generality.
Conclusion
In this thesis I have developed a simple approach to model elastic filaments. All equa-tions were derived starting from the basic classical mechanics equaequa-tions and all important steps were provided. Also, I have presented results from simulations that have validated the algorithm.
The algorithm considers a filament as a set of beads. These beads are constrained by three types of forces – stretching, bending and torsion. From the several contributions to the potential energy formulas the forces were derived, taking in account the intrinsic coordinates associated to each type of movement. An iterative method was then used to integrate the dynamical equation in order to obtain the filament movement along the time. All main details concerning software implementation were provided, and several simulations were performed to validate the algorithm. Several theoretical models were discussed and comparison of theoretical predictions and numerical results provided. I believe that this thesis shows that numerical integration of dynamical equations can indeed prove to be a valuable tool to study filament dynamics in a wide range of possible areas and applications.
Perspectives
The main objective of this thesis was to develop an algorithm that allows simulating Physical PseudoKnots (PPKs), Figure 3.6. PPKs are a new type of entanglement that are easily formed in helical structures. In polymer science, entanglements have been described according to the reptation model. In this model, the polymer entanglements constrain the movement of each chain within a tube. However, chains are still able to move and conse-quently the properties that this model predicts are rheological in nature. The formation of PPKs is purely physical and experiments suggest that it is a ratchet phenomenon. The creation of such entanglements produces very stable and localized links and hence it is ex-pected that they produce stable structural arrangements of polymers chains. It costs much more energy to destroy a PPK than to create it. In experiments with springs,a difference of almost two orders of magnitude between the creation and destruction energies of the PPKs was found.
PPKs have a probability to occur when two helical structures intertwine. With this simulation it is possible to identify the impact of different elastic (twisting, bending and stretching elastic constants) and geometrical properties (helical radius, thickness, etc.) on the probability of forming PPKs, and on their stability. It would be desirable to find general geometrical properties required to induce PPK formation in some structures, like answering the question why helices are special.
The algorithm developed in this thesis considers phantom chains. Effects of collisions and inter-crossing of chain segments are not taken into account. To make more realis-tic simulations of filaments, one can implement a repulsive potential like the potential described in section 1.4.
Appendix A
Torsional term derivation of the force
In the first chapter, I derived the different types of forces. The main strategy is quite simple but, in the case of the torsional force, the calculation and simplification of the force terms are not trivial. In this appendix, I will show the calculation of one of the four components of the torsional force. The other three are very similar to this one, so the other results can be calculated in the same way.
Using equation (1.19): Fi = − ∂Vtors ∂τi−2 ∂τi−2 ∂ri +∂V tors ∂τi−1 ∂τi−1 ∂ri +∂V tors ∂τi ∂τi ∂ri +∂V tors ∂τi+1 ∂τi+1 ∂ri = (A.1a) = − Ci−2∆τi−2 ∂τi−2 ∂ri + Ci−1∆τi−1 ∂τi−1 ∂ri + Ci∆τi ∂τi ∂ri + Ci+1∆τi+1 ∂τi+1 ∂ri . (A.1b)
The angle τj is given by
τj ≡ ∠ (ˆyj, ˆyj+1) = arccos (sj−1× sj) · (sj × sj+1) k (sj−1× sj) kk (sj × sj+1) k . (A.2) The derivative ∂τi−1 ∂rν i = − 1 sin τi−1 ∂ cos τi−1 ∂rν i (A.3a) = − 1 kpi−1× pik ∂ ∂rν i
(pi−1· pi) − cos τi−1
∂ ∂rν i (pi−1pi) . (A.3b) Calculating ∂ ∂rν i (pi−1· pi) = ∂ ∂rν i
(si−2· si−1) (si−1· si) − (si−2· si) (si−1) 2
(A.4a) = (si−1· si) sνi−2+ (si−2· si−1) −sνi−1+ s
ν
i + (si−1) 2
sνi−2− 2 (si−2· si) sνi−1
(A.4b)
and ∂
∂riν (pi−1pi) = pi
si−2sin βi−1
∂si−1 ∂rνi + si−2si−1 ∂ sin βi−1 ∂rνi + pi−1 sisin βi ∂si−1 ∂rνi + si−1sin βi ∂si ∂riν + si−1si ∂ sin βi ∂rνi (A.5a) = pi pi−1
(si−2)2sνi−1− si−2si−1cos βi−1sνi−2
+pi−1
pi
(si)2sνi−1− (si−1)2sνi − si−1sicos βi −sνi−1+ s ν i (A.5b) = pi pi−1 [pi−1× si−2] ν − pi−1 pi [pi× (si−1+ si)] ν , (A.5c)
we finally obtain the final expression of the previous derivative: ∂τi−1 ∂rν i = − 1 kpi−1× pik " (pi−1) 2
pi− pi−1picos τi−1pi−1
(pi−1)2
× si−2
− (pi)
2
pi−1− pi−1picos τi−1pi
(pi) 2 × (si−1+ si) #ν (A.6a) = − 1 ti−1 (pi−1× pi) × pi−1 (pi−1)2 × si−2+ (pi−1× pi) × pi (pi)2 × (si−1+ si) ν (A.6b) = − 1 ti−1 ti−1× ˆyi−1 pi−1 × si−2+ ti−1× ˆyi pi × (si−1+ si) ν . (A.6c)
Note that significant simplifications are made using relations of the kind A × (B × C) = B(A · C) − C(A · B).
Appendix B
Solution of one type of nested
radicals
In this Appendix, I will present the solution of the limit that appears in Chapter 2, but in a more general form
lim ϑ→∞2 ϑ v u u u u t2 − v u u u t2 + v u u t2 + ... + γϑ−3 s 2 + γϑ−2 r 2 + γϑ−1 q 2 + γϑ √ 2 = φπ, (B.1) with γi = −1, 0 or 1.
Starting from the combination of γi’s that gives the lower value, that is γi = γ = 1, the
nested radical has the solution s 2 + r 2 + q 2 +√2 + . . . = X =√2 + X . (B.2)
Solving this equation, we find the real solution X = 2. The limit we want to calculate has the indeterminate form ∞ × 0, because 2ϑ → ∞ and √2 − X → 0, when ϑ → ∞.
Numerically solving this limit, it can be seen that it tend to π.
Figure B (left) shows the values of the limit for other combinations of γi’s. If, instead
of symbols, we use colours (−1 = yellow, 0 = black and 1 = green), a fractal behaviour is obtained, Figure B (right).
φ γϑ−4 γϑ−3 γϑ−2 γϑ−1 γϑ 1 + + + + + 2 + + + + 0 3 + + + + -4 + + + 0 0 5 + + + - -6 + + + - 0 7 + + + - + 8 + + 0 0 0 9 + + - - + 10 + + - - 0 11 + + - - -12 + + - 0 0 13 + + - + -14 + + - + 0 15 + + - + + 16 + 0 0 0 0 + -9 8 7 6 5 4 3 2 1 0 500 1000 Ån-Λ ΦΠ
Figure B.1: (left) Limit results for other combinations of the present nested radical. In the (right) figure colours are used instead of symbols (−1 = yellow, 0 = black and 1 = green) a fractal behaviour is obtained, Figure B (right).
