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Honours Thesis Numerical Modelling of an Artificial Molecular Motor

By: Dain Hedgpeth 3234620 Thesis Supervisor: Paul Curmi Submitted 26th June 2008

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Index 1. Background And Introduction 1.1. Molecular Motors 1.2. The Inchworm Motor 1.3. Goal Of This Project 2. Methods 2.1. Lattice Models And Lattice Dynamics 2.2. The Polymer Simulation 3. Results And Discussion 3.1. Polymer Bound At One End 3.2. Polymer Bound At One End Constrained In A Tube 3.2.1. Tube Of Width 3 3.2.2. Tube Of Width 5 3.2.3. Tube Of Width 7 3.2.4. Tubes Of Width 9,10,11,15 & 41 3.3. Investigating Crossing Of X Axis At Various Tube Widths 3.3.1. Tube Of Width 3 3.3.2. Tube Of Width 5 3.3.3. Tube Of Width 7 3.3.4. Tube Of Width 9 3.3.5. Tubes Of Width 11,15,19,25,29,35 & 41 3.3.6. Cases Of End Node Traversing / Not Traversing Y Axis 3.4. Times Of First Passage Over Large Number Of Simulations 4. Conclusion 5. Proposals For Future Study 6. References

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Chapter 1: Background And Introduction 1.1 Molecular Motors Molecular motors are biological molecular machines that are the essential agents of movement in living organisms. Generally speaking, a motor may be defined as a device that consumes energy in one form and converts it into motion or mechanical work; for example, many protein-based molecular motors harness the chemical free energy released by the hydrolysis of ATP in order to perform mechanical work. Some examples of organically occurring molecular motors are the Myosin protein which is responsible for muscle contraction and Kinesin which moves cargo inside cells away from the nucles. In terms of energetic efficiency, these types of motors can be far superior to currently available man-made motors. [1] Additionally very little is understood about the internal processes in molecular motors and this makes it a field of great interest. Several research groups currently have plans to design and fabricate rudimentary man made molecular motors in a hope of learning more about how these devices operate in nature.

1.2 The Inchworm Motor The Inchworm Motor is one of several proposed molecular motors currently under consideration. The aim is to fabricate a device at nanometer scale which can translocate the mass of a given polymer using chemical energy from an external source. The proposed motor consists of a DNA polymer suspended in fluid and confined within a nanotube. In order to translocate the mass of the polymer along the tube axis, two processes will be utilized which can be triggered by introduction of chemical elements into the system. Ligand Facilitated Binding Along the length of the interior of the nanotube is attached random distribution of ligand-gated binding proteins denoted A and B in figures 1.2-1.8. The density of these sites is controllable in fabrication. Each end of the DNA has a target site for A and B, and in the presence of a one of two 3|Page

particular ligands in the surrounding fluid, an attractive force is experienced between a site and its relative protein encouraging it to bind and the polymer to become tethered. Controlled Extension And Compression of Chain As DNA has an inherently strong electrostatic charge along its length the molecule is strongly self avoiding, making it quite stiff with a long persistence length. If however free ionised electrons are present in the environment, they are attracted to and cluster around the positive charge of the DNA chain and screen the charge beyond a given distance. This greatly reduces the self-repulsive force between nearest neighbours in the chain and thus the apparent persistence length. The effect of selfavoidance between separated points of the polymer is also reduced and the DNA in this environment will tend to fold up into a more compact state, thus having a shorter mean end to end length. It will also become much more susceptible to bending induced by environmental fluctuations and currents. • •

Note that the concentration of ions in the environment surrounding the DNA polymer shall be denoted ‘ionic strength’. The ratio of end to end length to absolute length of a polymer shall be denoted ‘extension’.

FIG. 1.1 Log-log plot of -DNA extension as a function of ionic strength for the three channel widths used (Filled diamonds, circles, and squares TBE series measurements in channels of width 200, 100, and 50 nm respectively; open diamonds are a polymer unrelated to this project). (a) Fits of de Gennes scaling to extension measurements using the known ionic strength dependence of the effective width and persistence length. (b) The subpersistence length Odijk theory plotted against extension measurements (bold curve prediction for 50 nm channel, dashed curve prediction for 100 nm channel, and dotdashed curve prediction for 200 nm channel).The de Gennes theory gives the better agreement for all three curves. The error bars given on the extension are the standard deviations from the extension of all molecules measured in the channel at a given ionic strength, between 5 and 20 molecules for the 100 and 200 nm channels and between 2 and 5 molecules for the 50 nm channel.

Figure 1.1 shows results taken from reference [2] of a series of measurements of DNA extension at different ionic concentrations with various fittings that are beyond the scope of this discussion. Three series of results are overlaid; representing the data for a DNA molecule constrained in nanotubes of width 200, 100 and 500nm (points represented by empty diamonds should be ignored). The graph

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demonstrates that for all three different tube widths the measured, an increase in ionic strength corresponds to a decrease in DNA extension. The Inch-Worm Mechanism It is proposed that a repeated sequence of chemicals could be pumped through the nanotube in order to trigger the series of events displayed in figures 1.1-1.7.

Fig 1.2. The DNA polymer is bound by its ligands to an A site and a B site. The saline/ionic content is low and hence the polymer has a long end to end length.

Fig 1.3. The ligand which induces binding of B to its target site is removed. The end of the polymer is free to move but still tethered at end A.

Fig 1.4. The ionic content of the environment is increased. This causes shielding and the polymer folds up. It also moves and bends more in response to environmental turbulence and internal energy of the molecule.

Fig 1.5. The ligand which induces binding of B to its target site is reintroduced. The compressed polymer will now bind to a site further along the x axis than before.

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Fig 1.6. The ligand which induces binding of A to its target site is removed. The respective end of the polymer is now free to move.

Fig 1.7. The ionic strength is reduced, removing electrical shielding and causing the polymer to straighten.

Fig 1.8. The ligand which induces binding of A to its target site is reintroduced. The Inch Worm has now performed one complete walk cycle and has translocated in the negative x direction.

In this manner one effective cycle of a chemically powered molecular motor is completed. The chemicals could theoretically be cycled at a rapid rate through the nanotube causing repeated walk cycles to be completed and the polymer would continue to propagate along the tube axis. Current technology would be able to cycle these chemicals through the nanotube at a high rate, between nano and microsecond frequency.

Fabrication Constraints – Rectangular Cross Section of Nanotube Currently available fabrication processes will create nanotubes with rectangular cross sections. The possible dimensions include 62x44nm, 107nmx91nm and 232x173nm.

Possible Parameters To Be Explored And Optimized • Capture radius • Capture time • Track spacing • Chemical cycle timing • Effect of tube width on above and on flipping probability

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1.3 Goal Of Project This project begins preliminary modelling and analysis of the dynamics of certain components of an inchworm motor. A dynamic model of a polymer chain was developed as a computer program and subjected to a range of simulation conditions. In constructing a real inchworm motor certain parameters must be decided upon including tube dimensions, chemical pulse timing and concentration, polymer length, and binding site radius. These will affect properties such as the likelihood of finding a binding site, average time till binding, the distance travelled each walk cycle and probability of the polymers mass traversing to the other side of the bound point and hence changing the direction of propagation. All of these dynamical properties are defined by the polymers behaviour when it is in the state of being bound at one end only, where being bound means the end node of the chain is constrained to a fixed point on the lattice and may not move. These are the states depicted in figures 1.3, 1.4, 1.6 & 1.7. The model created for this thesis was constructed to simulate polymers bound at one end with the additional ability to be constrained within a given space representing the nanotube. It is hoped that these simulations and their analysis provide some useful qualitative insight into the behaviour of a real inchworm motor and some preliminary knowledge to help in determining the physical parameters required for its fabrication.

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Chapter 2: Methods 2.1 Lattice Models And Lattice Dynamics An inchworm device exists in one of two states; being tethered at one end or being tethered at noth ends. Being tethered refers to the end bound to a positional point and unable to move. Using the lattice model a series of computer simulations It was decided to develop and use a lattice dynamical polymer for the simulations. The lattice model is a method for greatly simplifying and approximating the underlying physics of a molecular simulation. It is a useful tool in polymer simulations and allows qualitative investigation of phenomena with greatly reduced computational time. Lattice Model Definition In the lattice model used the polymer was represented by a series of x and y coordinate pairs referred to as nodes, each resting at a discreet point on a periodic two dimensional lattice. The term node and x/y coordinate pairs may be used interchangeably. The line between a successive pair of nodes represents a segment of the polymer and the angles between two successive segments are constrained to be discreet multiples of 90°. The total set of coordinates which describe the positional state of a polymer are referred to as a conformation of the polymer and the total set of possible conformations of a polymer in a given simulation scenario is referred to as the conformation space. Additionally the total set of nodes representing a polymer in the lattice model will sometimes be referred to as a chain or the polymer chain. The model operates in discreet units of simulated time, which have no definite proportional relation to real time. Simulated time will be referred to simply as time. Each unit of time the model may alter the conformation of the polymer chain and the total collection of changes performed in a single time step is referred to as a move. The persistence length of a polymer of a given degree of stiffness is defined as the maximum length of polymer which can be observed for which that length will remain acceptably straight under given environmental conditions. As changes in direction in the lattice model can only occur at unit length intervals along the chain for the purposes of this model one unit length on the lattice will be regarded as approximately equal to the persistence length of the polymer.

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2.2 The Polymer Simulation The simulation was developed in the Python2.4 programming language. This language was chosen as it has built in functions which facilitate easy manipulation of files and of the required container classes. The Polymer Chain The polymer chain is represented by a linked list of variable pairs containing the nodes or positional coordinates of the polymer chain. The length of the chain is defined as N nodes. Move operations are performed by maintaining a linked list of integer variables ݊ from 0 to 3. Each variable represents the direction of the vector joining nodes ݊ and ݊ + 1 in the polymer chain, ሼ0,1,2,3ሽ = ሼ‫ݕ‬ො, ‫ݔ‬ො, −‫ݕ‬ො, −‫ݔ‬ොሽ. Moves are performed as a change to the values of these directional variables. The program then iterates along the variable chain and calculates the positional variables of each successive node, the first node lying on a predetermined origin. This method simplifies move calculations, allowing for easy manipulation of the conformation through integer changes in the required direction variable list. The Move Set Moves are chosen randomly with an equal probability weighting for each move. The set of possible moves is as follows –

Fig1.1.1. Set of Available Moves To be Performed On Polymer Chain

A.i) Corner flip – A right angle corner of on the polymer chain would have both vectors rotated by 90 degrees in the direction that preserved connection at a vertice. A.ii) End bead flip – if the angle between the last two nodes in the chain is 90 degrees, change it’s sign. B.i) Crankshaft rotation – A U-shaped section of the polymer would be inverted such that the curve of the U was in the opposite direction. B.ii) Solid rotation – the entire polymer beyond a given point would be rotated by 90 degrees.

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In all simulations for this project, moves were per performed by randomly selecting a node along the polymer chain and randomly applying a move from the move set. The weighting for each option was evenly distributed and a new random seed was used each time. Timestepping Each iteration of the program creates a copy of the polymer chain, performs a move from the move set on it and then tests that it satisfies the given acceptance conditions. If the move is accepted, the original polymer chain is replaced with the copy and the process is repeated. Acceptance Criteria: Self-avoidance The coordinates of each node in the proposed configuration are compared with one another and if more than one node lies at any one point on the lattice the configuration is rejected. Simulation of Physical boundary In cases where the boundary walls of a nanotube are simulated the coordinates of each node in the proposed configuration are observed and if any one point lies outside the tube width the configuration is rejected. Collected Data Data output which could be collected during simulations included: • Number of attempted conformations • Number of accepted conformations • Number of conformations rejected due to wall collision • Number of conformations rejected due to self-evasion • Two dimensional density plot of coordinates visited by the polymer • Coordinates of end node in polymer chain at each iteration • Time of occurrence for first crossing of the end node of the polymer, where first crossing refers to the first time the node crosses the y axis. • The conformation of the polymer at specified moments

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Chapter 3: Results & Discussion The aim was to simulate and investigate the behaviour of a polymer bound at one end and free to move in a nanotube. First however the simulation was first run with a simpler configuration such that a polymer was bound by one end to a point on the lattice with no additional constraints on movement of the chain. This would allow an initial test of the dynamics of the lattice model against expected behaviour.

3.1 Polymer Bound At One End Configuration Parameters The first simulation run intended to explore the simple situation of the simulated polymer bound at one end. The initial conformation of the chain in each instance was in a straight line along the x axis of the lattice. The only acceptance condition for a proposed move was that it be self-avoiding, there are no wall boundary conditions in this case. The steps performed in the simulation are such that the time taken to process each iteration is proportional to the number of nodes in the polymer. Due to time considerations, a polymer length of 38 nodes was decided upon as a reasonable length with which to perform the intended range of simulations. All results for this thesis were obtained using a 38mer. The first simulation was a run of 5x10^5 iterations performed with the polymer bound at (38,38) on the lattice. To analyse the path of the polymer over this time period an array of integer variables was stored in computer memory with the same dimensions as the lattice. For each iteration a given coordinate on the lattice was occupied by a polymer node, the corresponding coordinate on the array would be incremented by one. In this fashion a two dimensional density plot was created representing the amount of time part of the polymer spent on any one point of the lattice and this array has been processed in various forms in figures 3.1,3.2 and 3.3.

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Fig 3.1. Two dimensional density plot of molecule on lattice over 5x10^5 iterations. The x and y axes represent the physical dimensions of the lattice. For each iteration of the simulation a point on the lattice was occupied the corresponding value on the density plot was incremented by one. The value of each point is represented by colour, the lowest value being represented by dark green and the highest by dark red with a smooth gradient between. The image is interpolated to smooth out sharp edges created by the lattice structure. One end of the polymer chain is bound to point (38,38)

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198698 100000 98698 481.2

Moves attempted in total Moves accepted Attempts violated self-evasion Seconds to complete

Table 3.1. Statistics related to simulation run for figure 3.1 Tethered Chain 500,000 Iterations

Fig 3.2. Density plot of molecule on lattice over 5x10^5 iterations. An alternate view of the density plot from fig3.1. The x axis represents the x position on the lattice and the y axis represents the number of iterations for which a given coordinate was occupied, or its density of occupation. Contour lines run parallel to the x axis and are each one unit distance in the y direction from their neighbours. The bordering outline of the coloured shape represents the cross section of density along the x axis and is also the contour of highest points parallel to the x axis. 400000 350000 300000 250000 200000 150000 100000 50000 0 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 Fig 3.3. Density plot of molecule on lattice over 5x10^5 iterations integrated along y axis. The density of occupation of each point of the lattice is integrated across the y axis of the lattice. The x axis in the figure represents the x coordinate of the lattice while the y axis of thee figure represents the total density of occupation that coordinate of x.

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A full continuous electrostatic model in these circumstances would be free to move randomly around the bound point, thus it would be expected that over a large enough period of time it would create a circularly symmetric distribution. It would also be expected that this distribution attenuate with radial distance from the bound point as the closer a molecule is to the bound point the less area it has available to occupy and it thus spends more time on average over the points it visits. This behaviour is indeed as expected as demonstrated in the simplified lattice model fig3.1, with the density dropping off as a function radial distance from the tethered point at (38,38), sharply at first and then more gradually moving outwards. There is a high peak in the middle seen clearly in 3.2 and 3.3 at the lattice point (38,38). Visual examination of fig3.1 shows the distribution to be fairly symmetric around the tethered point also. So in this case the model seems to be qualitatively similar to its full dynamic counter-part. Interestingly the density vanishes well before the distance of the full polymer length along the radius, approaching zero at around 26 units from the central bound point. This indicates the polymer was spending the vast majority of its time in a compressed state. To investigate this further the distribution of position of the free end of the polymer was observed. Another simulation run of 5x10^5 iterations was performed and a density plot made of the position of the end node of over the polymer over the length of the run. 3D Sequential Graph (histogram41 79v*78c)

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Fig 3.4. Two dimensional density plot of position of end point of molecule on lattice over 5x10^5 iterations. The x and y axes represent the physical dimensions of the lattice. For each iteration of the simulation a point on the lattice was occupied by the end point of the chain, the corresponding value on the density plot was incremented by one. The value of each point is represented by colour, the lowest value being represented by dark green and the highest by dark red with a smooth gradient between. The image is interpolated at a low resolution smoothing out sharp edges. One end of the polymer chain is bound to point (38,38)

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1200 1000 800 600 400 200 0 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 Fig 3.5. Cross section along x axis of density plot of position of end point of molecule over5x10^5 iterations. The x axis in the figure represents position along the x axis of the lattice, while the y axis represents the density of occupation that coordinate of x by the end node in the polymer chain. Periodic zeros in density indicate possible characteristic behaviour of lattice model. 1200 1000

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Fig 3.6. Cross section along x axis of density plot of position of end point of molecule over5x10^5 iterations plotting even values of x only. This is the same format as fig3.5 but with density values at odd numbered x coordinates omitted to give a smooth distribution.

x coordinates most visited by end node along x axis Average distance from origin of end node of polymer along x axis

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Table 3.2. Data extracted from fig3.6. All values are taken from cross section along the x axis of the density distribution of the end node of the polymer over 5x10^5 iterations.

Figure 3.4 shows the distribution of the free end of the polymer over time, with a cross section of this distribution shown in fig3.5. The sharp periodic spikes along the y axis in fig3.5 were of interest and upon examination of the density plot data it seems that the end point has only visited alternating coordinate points of the lattice similar to the pattern of a chess board. As this pattern was strictly adhered to it is likely an inherent feature of the particular lattice dynamical model. In attempting to 14 | P a g e

investigate a close approximation to a full continuous dynamic model only the sites which were able to be visited were of interest, and the data for other points were removed to create the adjusted crosssection in fig3.6. Fig3.4 is radially symmetric and this would likely be expected as the end point is free to randomly explore the space within a given radius of the bound point. The only condition influencing the acceptance of moves is self-avoidance and this would have no bias in the radial direction. The density in both fig3.4 and fig3.6 starts at zero from an extreme radial distance then begins to increase at a distance of about 26 units from the bound centre at (38,38). The density peaks to around 1000 occurrences at the highest point of the cross section at around 10-12 units distance and then drops down to below 200 at the radial centre. The actual value at coordinate at the radial centre was removed in creating the adjusted cross-section profile in fig3.4 though due to self-avoidance the end node would not be able to visit the bound origin and this part of the density profile would likely drop to zero in a continuous model. As can be seen in table 3.2, which presents the most visited and average x coordinates along the x axis cross section, the polymer spends most of its time a little under a third of its absolute length. This would be logically explained in that straight and ordered states represent only a small proportion of the conformation space and would be less likely to occur. The most common length would most likely be the one with the most possible conformations and thus many bends in the polymer. Further study might test these lengths against theory by analysing the enumerated states of the system.

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3.2 Polymer Bound At One End Constrained In A Tube 3.2.1 Tube Width Of 3 Units Simulated Nanotube In the simplified model of the inchworm being simulated each walk cycle includes the transition from the state represented in figure 1.3 to that in 1.4. This shall be denoted the contraction step. Here the polymer begins completely straight along the x axis, tethered at one end then has it’s persistence length shortened and is allowed to curl up and convect under its own internal and environmental forces. Given the simulation tools developed thus far this was the first stage in the walk cycle to logically attempt to model. If the system were setup with a polymer running the length of a simulated tube, bound to a point along the tubes axis by one end and the initial conformation were with the chain completely straight along the tube axis, this would roughly approximate this stage of the walk cycle. This might assist in investigating qualitative ideas about values such as the rate of polymer contraction which would determine walk cycle frequency, optimum capture radius and capture point spacing. To simulate this a series of simulations were performed with the additional acceptance condition that the polymer avoid a spatial boundary representing the nanotube walls, representing the walls of a 2D nanotube. Simulations were run with wall widths of (3,5,7,8,9,10,15,20,25,30,35,40) all for of 5 × 10ହ iterations. Note that the term origin will be used from here on to refer to the location of the bound node of the polymer string and not the zero coordinate on the lattice.

Fig 3.7. Density plot of molecule constrained in tube of width 3 units over 5x10^5 iterations. The x axis represents the x position on the lattice and the y axis represents the number of iterations for which a given coordinate was occupied. The bordering outline of the coloured shape represents the cross section of density along the x axis and is also the contour of highest points parallel to the x axis.

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Fig 3.8. Two dimensional density plot of molecule constrained in tube of width 3 units over 5x10^5 iterations. The x and y axes represent the physical dimensions of the lattice. For each iteration of the simulation a point on the lattice was occupied the corresponding value on the density plot was incremented by one. The value of each point is represented by colour, the lowest value being represented by dark green and the highest by dark red with a smooth gradient between. The image is interpolated to smooth out sharp edges created by the lattice structure. One end of the polymer chain is bound to point (38,38)

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180000 160000 140000 120000 100000 80000 60000 40000 20000 0 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76

Fig 3.9. Density plot of molecule constrained in tube width 3 over 5x10^5 iterations integrated along y axis. The density of occupation of each point of the lattice is integrated across the y axis of the lattice. The x axis in the figure represents the x coordinate of the lattice while the y axis of thee figure represents the total density of occupation that coordinate of x.

Figure 3.8 shows the two dimensional density plot of the comformations of the molecule on the lattice over the time of simulation. Figure 3.7 shows a profile view of the density along the x axis, with the height of the coloured area tracing the cross section along the x axis. The graph is completely polarised, with all the density lying on the right hand side of the nanotube. There is a sharp peak in density around the bound end of the polymer seen clearly in 3.7. Similar to the previous simulation in figure 3.4 it is intuitive that there would be a higher density close to the bound origin of the polymer. The spike declines sharply to the right and levels out about 3-4 units away from the origin, remaining approximately level at 65-70% of the peak density before beginning to rapidly taper off at about 23 units from the origin and approaching zero at about 30 units distance. Fig3.9 which shows the total density at each x coordinate also shows this feature. Again the chain will likely spend the largest amount of time extended to the length which has the highest conformal degeneracy. The conformations of the polymer could be placed in two broad categories. The trail off in density from ~x-23> would thus be explained due to the decreasing probability of the chain being in a straight and ordered state. It would also be expected that this drop off become more pronounced with increasing tube width due to the increased number of conformations made available for shorter end to end lengths of the polymer.

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3.2.2 Tube Width Of 5 Units

Fig 3.10. Density plot of molecule constrained in tube of width 5 units over 5x10^5 iterations. The x axis represents the x position on the lattice and the y axis represents the number of iterations for which a given coordinate was occupied. Contour lines are parallel to the x axis and are spaced at one unit intervals in the y direction of the lattice. The bordering outline of the coloured shape represents the cross section of density along the x axis and is also the contour of highest points parallel to the x axis.

Fig 3.11. Two dimensional density plot of molecule constrained in tube of width 5 units over 5x10^5 iterations. The x and y axes represent the physical dimensions of the lattice. For each iteration of the simulation a point on the lattice was occupied the corresponding value on the density plot was incremented by one. Density value is

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represented by a colour gradient, the lowest being represented by dark green and the highest by dark red. The image is interpolated to smooth out sharp edges created by the lattice structure. One end of the polymer chain is bound to point (38,38)

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Fig 3.12. Density plot of molecule constrained in tube width 3 over 5x10^5 iterations integrated along y axis. The density of occupation of each point of the lattice is integrated across the y axis of the lattice. The x axis in the figure represents the x coordinate of the lattice while the y axis of thee figure represents the total density of occupation that coordinate of x.

A simulation run of 5x10^5 iterations was performed for a polymer in tube of width 5 and the results displayed in figures 10, 11 and 12 using the same analytic representations of the density distribution as figs 3.7,3.8 & 3.9 respectively. The distribution is similar in shape to the previous case but spread over the slightly wider tube width in the y direction. Two significant differences between the case of tube width 3 are observed. Firstly it is clear especially in fig3.12 that some portion of the polymer has spend time on the left hand side of the graph. The density on the left hand side of the origin sharply decreases to the right but reaches zero about 6 units to the left of the origin. The total area of this portion of the lattice would thus be about 30 (6 times the tube width), which contains less lattice points than nodes in the chain. Thus the entire polymer cannot have been on the right hand side of the tube at any one time, only a portion of it. A simplistic explanation of this phenomenon would be that to traverse the origin in such a thin tube would require movement of the mass of the polymer close to and past the origin. The self-avoidance condition causes this to be a very unlikely state, with the particle spending very little time folded over in such a dense conformation. This would result in a gradient in the density of the conformation space along the x axis, with a decrease in available states as the chain’s centre off mass crossed the origin and thus a tendency for the particle to remain on the right hand side.

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3.2.3 Tube Width Of 7 Units

Fig 3.13. Density plot of molecule constrained in tube of width 5 units over 5x10^5 iterations. The x axis represents the x position on the lattice and the y axis represents the number of iterations for which a given coordinate was occupied. Contour lines are parallel to the x axis and are spaced at one unit intervals in the y direction of the lattice. The bordering outline of the coloured shape represents the cross section of density along the x axis and is also the contour of highest points parallel to the x axis.

Fig 3.14. Two dimensional density plot of molecule constrained in tube of width 5 units over 5x10^5 iterations. The x and y axes represent the physical dimensions of the lattice. For each iteration of the simulation a point on the lattice was occupied the corresponding value on the density plot was incremented by one. Density value is

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represented by a colour gradient, the lowest being represented by dark green and the highest by dark red. The image is interpolated to smooth out sharp edges created by the lattice structure. One end of the polymer chain is bound to point (38,38) 300000 250000 200000 150000 100000 50000 0 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76

Fig 3.15. Density plot of molecule constrained in tube width 3 over 5x10^5 iterations integrated along y axis. The density of occupation of each point of the lattice is integrated across the y axis of the lattice. The x axis in the figure represents the x coordinate of the lattice while the y axis of thee figure represents the total density of occupation that coordinate of x.

First Crossing of Bulk of Chain Figures 3.13,3.14 & 3.15 represent the density distributions for a simulation of 5x10^5 iterations again presented respectively in the same analytic forms as previously. In 3.13 moving along the x axis from the left there is now a moderate rise about 23 units from the origin which curves to a plateau at 10 units distance and levels off completely at 3 units before sharply rising to a peak at the origin. On the right hand side of 3.13 moving in towards the origin there is a similar shape though reflected horizontally around the origin. The density starts to rise at 23 units from the origin, begins to taper off around 22-23 units, plateaus at about 10% of the peak density and begins to rise somewhat more gradually from about 8 units out towards the origin. With the distribution heavily polarised now on the opposite side of the graph, this implies that the bulk of the polymer at some point managed to cross and spend more time on the left hand side than the right. It is clear that with this tube width there is a finite probability of the polymer being able to cross the tube length though from this information it is undetermined whether for smaller widths a crossing could occur given the constrained space of such a small area of the lattice and available move set. Since there is a finite probability of crossing it is expected that given enough simulation time the system would equilibrate on both sides of the tube and the density distribution would approach complete symmetry over time. It was hypothesised at this point that further widening of the tube would have a similar effect, as the increased width would reduce the constriction of conformation space around the origin and the bulk of the polymer would be more likely to cross the length of the tube in a given time. So for increasing tube widths we would expect to see the system moving closer to equilibrium and symmetry on the x axis around the origin for the same number of simulation iterations.

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Additionally further rounding of the drop off in the probability plateau extending from the origin is also seen as the configuration space increases and the density extending out from the origin is spread over a larger area.

For Possible Further Thought and Study –Radial And Symmetric Distribution If on a continuous dynamic model the density were taken purely as a function of radial distance, ie. Taking the integral 360degrees around the origin in circular coordinates at each radial distance in the lattice space the following might be predicted: •

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For tube width approaching 1 the conformation space is reduced to 1 where the polymer has only one possible length and the density as function of radius would approach a step function with finite value from the origin to the absolute length of the chain. For tube width approaching the absolute length of the polymer the conformation space would be increased and the most common conformations would become those which are more curved and shorter. Thus the density would have its largest contributions near the centre and extending out for a distance before beginning to taper off to zero. This would describe a step function with a tapered end as described in Table 3.3.

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Step function. Tapered step function with rounded curve downwards heading away from origin.

Table 3.3

3.2.4 Tube Widths Of 9 – 41 Units Further series of simulations were done, all for 5x10^5 iterations and at tube widths of between 9 and 41 units. Again the density plots are analysed and presented in the same respective forms as previously.

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Fig 3.16. Density plot of molecule constrained in tube of width 5 units over 5x10^5 iterations. The x axis represents the x position on the lattice and the y axis represents the number of iterations for which a given coordinate was occupied. Contour lines are parallel to the x axis and are spaced at one unit intervals in the y direction of the lattice. The bordering outline of the coloured shape represents the cross section of density along the x axis and is also the contour of highest points parallel to the x axis.

Fig 3.17. Two dimensional density plot of molecule constrained in tube of width 5 units over 5x10^5 iterations. The x and y axes represent the physical dimensions of the lattice. For each iteration of the simulation a point on the lattice was occupied the corresponding value on the density plot was incremented by one. Density value is represented by a colour gradient, the lowest being represented by dark green and the highest by dark red. The

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image is interpolated to smooth out sharp edges created by the lattice structure. One end of the polymer chain is bound to point (38,38)

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Fig 3.18. Density plot of molecule constrained in tube width 3 over 5x10^5 iterations integrated along y axis. The density of occupation of each point of the lattice is integrated across the y axis of the lattice. The x axis in the figure represents the x coordinate of the lattice while the y axis of thee figure represents the total density of occupation that coordinate of x.

For Further Study - Increased Symmetry With a tube width of 9 the distribution over the same number of iterations has an increased level of symmetry about the origin on the x axis. Additional analysis might investigate the relationship between tube width and the time taken to reach a level of symmetry.

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Fig 3.19. Density plot of molecule constrained in tube of width 5 units over 5x10^5 iterations. The x axis represents the x position on the lattice and the y axis represents the number of iterations for which a given coordinate was occupied. Contour lines are parallel to the x axis and are spaced at one unit intervals in the y direction of the lattice. The bordering outline of the coloured shape represents the cross section of density along the x axis and is also the contour of highest points parallel to the x axis.

Fig 3.20. Two dimensional density plot of molecule constrained in tube of width 5 units over 5x10^5 iterations. The x and y axes represent the physical dimensions of the lattice. For each iteration of the simulation a point on the lattice was occupied the corresponding value on the density plot was incremented by one. Density value is represented by a colour gradient, the lowest being represented by dark green and the highest by dark red. The

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image is interpolated to smooth out sharp edges created by the lattice structure. One end of the polymer chain is bound to point (38,38) 300000 250000 200000 150000 100000 50000 0 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76

Fig 3.21. Density plot of molecule constrained in tube width 3 over 5x10^5 iterations integrated along y axis. The density of occupation of each point of the lattice is integrated across the y axis of the lattice. The x axis in the figure represents the x coordinate of the lattice while the y axis of thee figure represents the total density of occupation that coordinate of x.

Fig 3.22. Density plot of molecule constrained in tube of width 5 units over 5x10^5 iterations. The x axis represents the x position on the lattice and the y axis represents the number of iterations for which a given coordinate was occupied. Contour lines are parallel to the x axis and are spaced at one unit intervals in the y direction of the lattice. The bordering outline of the coloured shape represents the cross section of density along the x axis and is also the contour of highest points parallel to the x axis.

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Fig 3.23. Two dimensional density plot of molecule constrained in tube of width 5 units over 5x10^5 iterations. The x and y axes represent the physical dimensions of the lattice. For each iteration of the simulation a point on the lattice was occupied the corresponding value on the density plot was incremented by one. Density value is represented by a colour gradient, the lowest being represented by dark green and the highest by dark red. The image is interpolated to smooth out sharp edges created by the lattice structure. One end of the polymer chain is bound to point (38,38) 300000 250000 200000 150000 100000 50000 0 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76

Fig 3.24. Density plot of molecule constrained in tube width 3 over 5x10^5 iterations integrated along y axis. The density of occupation of each point of the lattice is integrated across the y axis of the lattice. The x axis in the figure represents the x coordinate of the lattice while the y axis of thee figure represents the total density of occupation that coordinate of x.

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Fig 3.25. Density plot of molecule constrained in tube of width 5 units over 5x10^5 iterations. The x axis represents the x position on the lattice and the y axis represents the number of iterations for which a given coordinate was occupied. Contour lines are parallel to the x axis and are spaced at one unit intervals in the y direction of the lattice. The bordering outline of the coloured shape represents the cross section of density along the x axis and is also the contour of highest points parallel to the x axis.

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Fig 3.26. Two dimensional density plot of molecule constrained in tube of width 5 units over 5x10^5 iterations. The x and y axes represent the physical dimensions of the lattice. For each iteration of the simulation a point on the lattice was occupied the corresponding value on the density plot was incremented by one. Density value is represented by a colour gradient, the lowest being represented by dark green and the highest by dark red. The image is interpolated to smooth out sharp edges created by the lattice structure. One end of the polymer chain is bound to point (38,38) 350000 300000 250000 200000 150000 100000 50000 0 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76

Fig 3.27. Density plot of molecule constrained in tube width 3 over 5x10^5 iterations integrated along y axis. The density of occupation of each point of the lattice is integrated across the y axis of the lattice. The x axis in the figure represents the x coordinate of the lattice while the y axis of thee figure represents the total density of occupation that coordinate of x.

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Fig 3.28. Density plot of molecule constrained in tube of width 5 units over 5x10^5 iterations. The x axis represents the x position on the lattice and the y axis represents the number of iterations for which a given coordinate was occupied. Contour lines are parallel to the x axis and are spaced at one unit intervals in the y direction of the lattice. The bordering outline of the coloured shape represents the cross section of density along the x axis and is also the contour of highest points parallel to the x axis.

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Fig 3.29. Two dimensional density plot of molecule constrained in tube of width 5 units over 5x10^5 iterations. The x and y axes represent the physical dimensions of the lattice. For each iteration of the simulation a point on the lattice was occupied the corresponding value on the density plot was incremented by one. Density value is represented by a colour gradient, the lowest being represented by dark green and the highest by dark red. The image is interpolated to smooth out sharp edges created by the lattice structure. One end of the polymer chain is bound to point (38,38) 350000 300000 250000 200000 150000 100000 50000 0 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76

Fig 3.30. Density plot of molecule constrained in tube width 3 over 5x10^5 iterations integrated along y axis. The density of occupation of each point of the lattice is integrated across the y axis of the lattice. The x axis in the figure represents the x coordinate of the lattice while the y axis of thee figure represents the total density of occupation that coordinate of x.

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400000 350000 300000 250000 200000 150000 100000 50000 0 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76

Fig 3.31. Density plot of molecule constrained in tube width 3 over 5x10^5 iterations integrated along y axis. The density of occupation of each point of the lattice is integrated across the y axis of the lattice. The x axis in the figure represents the x coordinate of the lattice while the y axis of thee figure represents the total density of occupation that coordinate of x.

As the tube width is made progressively wider two important features of the density distributions continue to change. Firstly the early rise and plateau of the density on each side of the origin moving inwards became progressively less accentuated, the distribution approaching the morphology of a smooth curve which begins to rise gradually at about 20 units from the origin and increases to rise more rapidly moving inwards to the sharp peak density at the origin, as in fig3.31. This profile is approaching the profile of the case with no wall interaction as in fig3.2 and this intuitively makes sense as less and less of the conformations would interact with the walls as the tube widened. The distribution can also be seen to approach circular symmetry with tube widening and is fairly close to this in fig3.29

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3.3 Investigating crossing of x axis at various tube widths It is apparent from section 3.2 that there is a possible relationship between the width of the nanotube and the probability of the polymer crossing the tube length. Keeping in mind that this sort of crossing in the case of the inchworm would cause a reversal of the direction of propagation, an undesired effect, it was aimed to investigate this further. It was aimed to investigate whether there was a specific tube width for which the polymer was unable to traverse the axis. If this were the case it would be useful in determining the scale of the tube width in relation to the persistence length of the polymer at which the inchworm would be certain to continue propagating in a given direction. See figures 1.21.7. A set of simulations were run for 5 × 10ହ iterations each with various tube widths from 5-41, this time recording the path of the end node of the polymer and its density distribution over time.

3.3.1 Tube Of Width 3

Fig 3.32. Two dimensional density plot of end node of molecule constrained in tube of width 3 units over 5x10^5 iterations. The x and y axes represent the physical dimensions of the lattice. Density value is represented by a colour gradient, the lowest being represented by dark green and the highest by dark red. One end of the polymer chain is bound to point (38,38)

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Fig 3.33. Two dimensional density plot of end node of molecule constrained in tube of width 3 units over 5x10^5 iterations. This figure presents the same data and format as fig3.33, except that the image is interpolated to smooth out sharp edges created by the lattice structure.

Similar to in figure 3.5, the density distribution of end point position in fig 3.32 has finite value only on alternate integer values of lattice coordinates, resembling a chess board. As it is hoped here to gain an understanding of the real world continuous model, an interpolated version of each two dimensional density plot in this section will be presented as the chess board effect is almost certainly an artefact of the lattice model and a smoothly graded distribution it is believed to be a closer approximation to the real physical dynamics. With the tube of width 3 the density of the end node distribution is concentrated in a fairly localized vicinity. As seen in the interpolated plot of density in 3.33, the density starts to rise rapidly from zero at 18 units from the origin, rising to a peak at 25 and falling sharply at 32 units distance. As the end point density remains entirely within a certain vicinity removed from the origin at positive x value, it is certain that the polymer did not cross the length of the tube at any time in this simulation. Due to self avoidance and a greatly constrained conformation space it seems that there would be a very strong tendency against the bulk of the polymer moving towards the origin. It is unclear however whether it would be possible for the chain to cross or if it is merely highly improbable. One conceivable passage might involve the bulk of the length of the polymer moving to the top (+y) of the tube and the end curling round and sliding past underneath, passing under the bound origin and taking the remainder of the chain with it. This however is not only a specific required conformational state, but a series of perhaps ~38 sequential conformational states required for this to occur. The probability of the polymer being in one particular state at any time is (though wall conditions and selfavoidance would likely decrease this by several orders of magnitude) in the order of 1 in 3^37. Thus the chance of a specific sequential series of states of length 38 occurring would likely be within a number of orders of magnitude of one in 3^(37^38). Further study might find theoretical enumerations of possible paths of crossing and more precise probabilities for given tube widths.

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3.3.2 Tube Of Width 5 Another simulation was run tracing the density of the end point of the polymer with a tube width of 5.

Fig 3.34. Two dimensional density plot of end node of molecule constrained in tube of width 5 units over 5x10^5 iterations. The x and y axes represent the physical dimensions of the lattice. Density value is represented by a colour gradient, the lowest being represented by dark green and the highest by dark red. One end of the polymer chain is bound to point (38,38) Tube Width 5

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Fig 3.35. Two dimensional density plot of end node of molecule constrained in tube of width 5 units over 5x10^5 iterations. This figure presents the same data and format as fig3.33, except that the image is interpolated to smooth out sharp edges created by the lattice structure.

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With the tube width at 5 units the behaviour is similar, with the distribution spread over a greater range of y and x as seen in fig 3.35 which plots end node density. Heading in the positive direction along the x axis of fig3.35 the density begins to rise at a lower value, having a very small value at x=~5, slowly rising and then curving upwards to the peak density at about 18. Though the minima visited location by the end node and the peak of its distribution are now shifted closer to the origin, the end point has still at no time crossed the x origin.

3.3.1 Tube Of Width 7 Another simulation was run tracing the density of the end point of the polymer with a tube width of 7. Tube Width 7

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Fig 3.36. Two dimensional density plot of end node of molecule constrained in tube of width 7 units over 5x10^5 iterations. The x and y axes represent the physical dimensions of the lattice. Density value is represented by a colour gradient, the lowest being represented by dark green and the highest by dark red. One end of the polymer chain is bound to point (38,38)

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Fig 3.37. Two dimensional density plot of end node of molecule constrained in tube of width 7 units over 5x10^5 iterations. This figure presents the same data and format as fig3.33, except that the image is interpolated to smooth out sharp edges created by the lattice structure.

Figure 3.37 is a plot of end point density for a simulation with tube width 7. In this case again the bulk of the distribution is spread yet further in both the x and y directions with a broader peak and more shallow slope to zero approaching the origin. Travelling right across the plot the density has a nonzero value first at 39, one unit away from the origin. So the polymer has not at any point here crossed to the left hand side of the tube.

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3.3.4 Tube Of Width 9 Another simulation was run tracing the density of the end point of the polymer with a tube width of 9.

Fig 3.38. Two dimensional density plot of end node of molecule constrained in tube of width 9 units over 5x10^5 iterations. The x and y axes represent the physical dimensions of the lattice. Density value is represented by a colour gradient, the lowest being represented by dark green and the highest by dark red. One end of the polymer chain is bound to point (38,38)

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Fig 3.39. Two dimensional density plot of end node of molecule constrained in tube of width 9 units over 5x10^5 iterations. This figure presents the same data and format as fig3.33, except that the image is interpolated to smooth out sharp edges created by the lattice structure.

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Figure 3.39 shows the end point density plot of the polymer in a tube width 9. Here there are two almost symmetrical density concentrations about the bound origin in the x direction. They compose of peaks roughly rounded rectangular in shape, ~10 units in width and ~15 units in length centred roughly around x=+-16 which have a shallow decrease in density towards the axis and comparably steep decrease away from it. The distribution appears to go to zero in the vicinity of the origin but retains some small finite value passing just above and below it. It is clear from this bipolar distribution that the chain has as some point crossed the length of the tube to the left hand side. The small nonzero densities above and below the bound origin would likely correspond to where the end point spent a brief amount of time crossing over the length of the tube. The fact that the two peaks are of comparable height indicates that a roughly equivalent amount of time was spent on either side. It would likely be hasty however to conclude that the system had a roughly symmetric distribution due to it becoming equilibrated as the next thinnest tube width simulation did not produce any crossing at all. It would be prudent to investigate these measurements over a large number of runs and gather a statistical distribution of the amount of time taken to cross and balance the density in order to explore this.

3.3.5 Tubes Of Width 11,15,19,25,29,35 & 41 A series of simulations were run tracing the density of the end point of the polymer with range of tube widths from 11-41 units.

Fig 3.40. Two dimensional density plot of end node of molecule constrained in tube of width 11 units over 5x10^5 iterations. The x and y axes represent the physical dimensions of the lattice. Density value is represented by a colour gradient, the lowest being represented by dark green and the highest by dark red. One end of the polymer chain is bound to point (38,38)

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Fig 3.41. Two dimensional density plot of end node of molecule constrained in tube of width 11 units over 5x10^5 iterations. This figure presents the same data and format as fig3.33, except that the image is interpolated to smooth out sharp edges created by the lattice structure.

Fig 3.42. Two dimensional density plot of end node of molecule constrained in tube of width 15 units over 5x10^5 iterations. The x and y axes represent the physical dimensions of the lattice. Density value is represented by a colour gradient, the lowest being represented by dark green and the highest by dark red. One end of the polymer chain is bound to point (38,38)

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Fig 3.43. Two dimensional density plot of end node of molecule constrained in tube of width 15 units over 5x10^5 iterations. This figure presents the same data and format as fig3.33, except that the image is interpolated to smooth out sharp edges created by the lattice structure.

Fig 3.44. Two dimensional density plot of end node of molecule constrained in tube of width 19 units over 5x10^5 iterations. The x and y axes represent the physical dimensions of the lattice. Density value is represented by a colour gradient, the lowest being represented by dark green and the highest by dark red. One end of the polymer chain is bound to point (38,38)

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Fig 3.45. Two dimensional density plot of end node of molecule constrained in tube of width 19 units over 5x10^5 iterations. This figure presents the same data and format as fig3.33, except that the image is interpolated to smooth out sharp edges created by the lattice structure.

Fig 3.46. Two dimensional density plot of end node of molecule constrained in tube of width 25 units over 5x10^5 iterations. The x and y axes represent the physical dimensions of the lattice. Density value is represented by a colour gradient, the lowest being represented by dark green and the highest by dark red. One end of the polymer chain is bound to point (38,38)

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Fig 3.47. Two dimensional density plot of end node of molecule constrained in tube of width 25 units over 5x10^5 iterations. This figure presents the same data and format as fig3.33, except that the image is interpolated to smooth out sharp edges created by the lattice structure.

Fig 3.48. Two dimensional density plot of end node of molecule constrained in tube of width 29 units over 5x10^5 iterations. The x and y axes represent the physical dimensions of the lattice. Density value is represented by a colour gradient, the lowest being represented by dark green and the highest by dark red. One end of the polymer chain is bound to point (38,38)

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3D Sequential Graph (histogram29 79v*78c)

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Fig 3.49. Two dimensional density plot of end node of molecule constrained in tube of width 29 units over 5x10^5 iterations. This figure presents the same data and format as fig3.33, except that the image is interpolated to smooth out sharp edges created by the lattice structure.

Fig 3.50. Two dimensional density plot of end node of molecule constrained in tube of width 35 units over 5x10^5 iterations. The x and y axes represent the physical dimensions of the lattice. Density value is represented by a colour gradient, the lowest being represented by dark green and the highest by dark red. One end of the polymer chain is bound to point (38,38)

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3D Sequential Graph (histogram35 79v*78c)

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Fig 3.51. Two dimensional density plot of end node of molecule constrained in tube of width 35 units over 5x10^5 iterations. This figure presents the same data and format as fig3.33, except that the image is interpolated to smooth out sharp edges created by the lattice structure.

Fig 3.52. Two dimensional density plot of end node of molecule constrained in tube of width 41 units over 5x10^5 iterations. In this instance the tube is wide enough such that no interaction is experienced between its walls and the molecule. The x and y axes represent the physical dimensions of the lattice. Density value is represented by a colour gradient, the lowest being represented by dark green and the highest by dark red. One end of the polymer chain is bound to point (38,38).

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3D Sequential Graph (histogram41 79v*78c)

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Fig 3.53. Two dimensional density plot of end node of molecule constrained in tube of width 41 units over 5x10^5 iterations. In this instance the tube is wide enough such that no interaction is experienced between its walls and the molecule. This figure presents the same data and format as fig3.33, except that the image is interpolated to smooth out sharp edges created by the lattice structure.

With increasing tube width seen in figures 3.40-3.53, the two characteristic peaks of the distribution could be seen to further broaden in the x and y directions and their max points move slightly towards the origin. The low density nonzero strips above and below the bound origin became increasingly dense and broad, moving slight away from the origin and approaching similar dimension and density of the two main peaks. The distribution eventually approached the case of a simulation with no tube walls which was circularly symmetric around the bound origin as in fig3.53. During the simulations and upon accidentally misplacing the results running a second series of identical simulations it was found that the polymer was able to completely cross the x origin in the tube width of 7 case. This raised doubts about the notion that a discreet cut off width existed for which a traversal could occur. Instead this seems to reinforce the idea that there might be a certain probability of traversal occurring within a given number of steps which is a function of tube width, where this probability becomes increasingly small as the width approaches 1.

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3.3.6 Cases of End Node Traversing / Not Traversing Y Axis In order to investigate the cases of polymer crossing and not crossing the length of the tube of width 7 more fully a more detailed analysis of the data sets was made. Case of Polymer Crossing Tube Length: Density Plots At Progressive Time Intervals The stored trajectory of the end node was decomposed into segments representing certain intervals of time. From this a plot could be made of the density distribution for each time period, allowing a more detailed view of the passage of the end node. Again the molecule is tethered at (x,y)=(38,38) and this is referred to as the origin. The first case observed was that in which the end point crossed the length of the tube at some point.

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(i) Fig 3.54. Density plots along x direction of tube of end node of polymer for case where end node crosses to other side of tube. Plots represent density at various x coordinates on the lattice between certain time intervals ranging from 10^4 to 10^5 iteration steps, the tube width is 7 units. X axis represents x coordinate on the lattice centred at the bound point of the polymer. Height on y axis represents occupation density of corresponding x coordinate between the start and finish of each time interval.

Sequence Of Events Figures 3.54a-i represent the end node density along the x direction for the polymer in a tube of width 7 at successive time intervals of length 10^4 iterations. In (a) the distribution comprises a peak centred around ~13 which decreases to zero at about x=+5 heading towards the origin. The polymer thus for this time frame remains entirely on the right hand side of the tube. In (b) the distribution comprises two distinct peaks of roughly equal height and joined by a strip of low density which runs across the origin. One is on the right of the origin centred around x=~9 and the other on the left hand side and centred around x=~-6. It is clear that the end node has crossed over to the other side of the tube but as the peak of the density and thus the polymers mean end to end length for this time period is roughly one half the mean end to end length for the first time period. It could be interpreted from this that perhaps the particle had reached a compacted state on the right hand side of the origin and the end node had crossed over taking a small amount of the chain with it and moving around somewhat. The bulk on the right hand side might still have been fairly constrained by its own compacted bulk and the origin and so the particle did not completely cross over. In (c) there are two distinct peaks, one centred around 13 and the other around the origin. If the previous hypothesis for (b) were taken to be true this be interpreted that the end point which had been able to cross over the left hand side of the origin a small distance was still held from passing through any further by the bulk on the right. It could be posed that the compacted bulk’s self repulsive force of self avoidance causing it to expand back out away from itself and the bound origin point caused the molecule to pull the free polymer end back through to the right hand side of the tube and the whole molecule again spent some time partially extended to the right. (d) and (e) show two distinct peaks centred at +-~13 and one distinct peak centred at -13 respectively. In (e) the entire distribution resides on the left hand side of the origin and it seems that between 30000 40000 iterations the end point of the polymer has crossed the tube length, while by 50000 iterations the entire polymer has certainly crossed over.

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(j) Fig 3.55. Cumulative density plots along x direction of tube of end node of polymer for case where end node crosses to other side of tube. The tube width is 7 units. X axis represents x coordinate on the lattice centred at the bound point of the polymer. Height on y axis represents occupation density of corresponding x coordinate from the start of the simulation until a given time between 10^4 and 10^5 iterations.

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Fig 3.56. Cumulative density plots of end node of polymer over laid upon each other for case where end node crosses to other side of tube. Plots shown are at time intervals of 10^5 from 0 to 10^6 iterations, the tube width is 7 units. X axis represents x coordinate on the lattice centred at the bound point of the polymer. Height on y axis represents occupation density of corresponding x coordinate from the start of the simulation until a given time.

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For further comparison figures 3.55 (a)-(j) provides density plots along the x axis for the same data and time intervals which is cumulative. Interesting to note was that the end point has apparently crossed over the origin a number of times: • • •

2000-3000 – The end point has crossed over and spent time on the left hand side of the origin, but does not travel far, spending the most time around 32‫ݔ‬ො, only 9 units from the origin 3000-4000 – The end point has spent some time near the origin and then crossed back to the right hand side 4000-5000 – The end point has crossed back to the left hand side of the origin and remains there

Case of Polymer Crossing Tube Length: Density Plots At Progressive Time Intervals Next is observed the case where the end node did not cross over the length of the tube at any point in the simulation. 18000 16000 14000 12000

10^6

10000

5x10^5

8000

10^5

6000

5x10^4

4000

10^4

2000 0 1

11

21

31

41

51

61

71

81

Fig 3.57. Cumulative density plots of end node of polymer over laid upon each other for case where end node does not cross to other side of tube. Plots shown are at time intervals from 10^4 to 10^6 iterations, the tube width is 7 units. X axis represents x coordinate on the lattice centred at the bound point of the polymer. Height on y axis represents occupation density of corresponding x coordinate from the start of the simulation until a given time.

Figure 3.57 shows the cumulative build up of density distribution over varying intervals from 0 to 10^6 iterations. It can be seen that density distribution corresponding to the final recorded time has no value on the negative side of the x axis. This indicates that over the whole time of simulation the end node did not cross or spend any time on the negative side of the x axis. This was somewhat surprising to have such wide ranging results, the first simulation run crossing three times in a given period and the second case not crossing at all in more than an order of magnitude longer. The number of crossings might be explained to an extend by the sequence of events posed for the former case, where the chance of the second crossing would be reduced due to the bulk of the polymer remaining on the opposing side. Nevertheless this is a considerable statistical spread.

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Adendum to 3.3.6 - Conformation At Time Of First Crossing The conformation of the polymer at the moment of traversal for the simulation which experienced crossing was recorded and is graphed below.

4 3 2 1 0 -2

0

2

4

6

8

10

-1 -2 -3 Fig 3.58. Conformation of polymer chain at moment just before end point crosses y axis. Tube width of 7 units

4 3 2 1 0 -2

0

2

4

6

8

10

-1 -2 -3 Fig 3.59. Conformation of polymer chain at moment just after end point crosses y axis. Tube width of 7 units

The bulk of the polymer at time of crossing was indeed in this case still on the right hand side of the origin as hypothesised, though without thorough theoretical understanding of the dynamical tendencies of the system the hypothesis cannot be decided upon with absolute certainty and should provide only a qualitative guide of possible interaction.

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3.4 Times Of Traversal Over Large Number Of Simulations Given the unpredictable nature of the time of first crossing of the end node discovered in part 3.3, it was decided to further investigate crossing times at various tube widths and see if a general relationship were apparent. It was aimed to perform a large number of tailored simulations who’s task was to simulate the tethered polymer in the nanotube only until crossing occurred then output the time of occurrence. The program code was altered to perform the following series of tasks: 1. Iterate the polymer until the end node traverses the y axis 2. Record the iteration number at which traversal occurred 3. Terminate iteration, reinitialize all variables and repeat from step 1 Note: As the stage of transition in the inchworm walk cycle from the state in fig1.3 to the state in 1.4 is being modelled it is important that all program variables are reinitialized before reinitiating the simulation so that the polymer returns to its straightened state along the tube axis. This ensures the times to crossing represent roughly the possible occurrences in a stage of a real inchworm motor. A large number of 500 – 20000 traversal times was collected for each tube width. This data was plotted as histograms of time of traversal vs. Number of occurrences.

Histogram (Tube width 5, 611 runs) 18 16

No of occurrences

14 12 10 8 6 4 2

(a)

2.6596E6

2.4701E6

2.2806E6

2.091E6

1.9015E6

1.712E6

1.5225E6

1.333E6

1.1434E6

9.5391E5

7.6439E5

5.7487E5

3.8535E5

1.9583E5

6306

0

Iterations Till Crossing Time

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(c)

27867.5765

25879.0353

23890.4941

21901.9529

19913.4118

17924.8706

15936.3294

13947.7882

11959.2471

(b)

9970.7059

7982.1647

5993.6235

4005.0824

2016.5412

28.0000

No of occurrences

1.9515E5

1.8121E5

1.6728E5

1.5335E5

1.3941E5

1.2548E5

1.1155E5

97613.4

83680.2

69747.0

55813.8

41880.6

27947.4

14014.2

81.0

No of occurrences

Var1 = 5830*774.0667*expon(x, 4.0202E-5)

Histogram (Tube width 7, 5831 runs)

200

180

160

140

120

100

80

60

40

20

0

Iterations Till Crossing Time

180 Histogram (Tube width 8, 5000 runs)

160

140

120

100

80

60

40

20

0

Iterations Till Crossing Time

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Histogram (Tube width 9, 10000 runs) 350 300

No of occurrences

250 200 150 100 50

25386.1882

23575.3176

21764.4471

19953.5765

18142.7059

16331.8353

14520.9647

12710.0941

10899.2235

9088.3529

7277.4824

5466.6118

3655.7412

1844.8706

34.0000

0

Iterations Till Crossing Time

(d)

Histogram (Tube width 10, 20000 runs) 1600 1400

No of occurrences

1200 1000 800 600 400 200 0 17.00 787.14 1557.28 2327.42 3097.56 3867.70 4637.84 5407.98 402.07 1172.21 1942.35 2712.49 3482.63 4252.77 5022.91

(e)

Iterations Till Crossing Time

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(g)

723.4235

671.9647

620.5059

569.0471

517.5882

466.1294

414.6706

363.2118

311.7529

(f)

260.2941

208.8353

157.3765

105.9176

54.4588

3.0000

No of occurrences

1752.2588

1627.8118

1503.3647

1378.9176

1254.4706

1130.0235

1005.5765

881.1294

756.6824

632.2353

507.7882

383.3412

258.8941

134.4471

10.0000

No of occurrences

700 Histogram (Tube width 15, 20000 runs)

600

500

400

300

200

100

0

Iterations Till Crossing Time

1800 Histogram (Tube width 20, 50000 runs)

1600

1400

1200

1000

800

600

400

200

0

Iterations Till Crossing Time

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Histogram (Tube width 30, 50000 runs) 2600 2400 2200

No of occurrences

2000 1800 1600 1400 1200 1000 800 600 400 200

284.6353

264.4471

244.2588

224.0706

203.8824

183.6941

163.5059

143.3176

123.1294

102.9412

82.7529

62.5647

42.3765

22.1882

2.0000

0

Iterations Till Crossing Time

(h)

At a tube width of 30, certain numbers of iterations had a particularly large number of occurrences. This could be a particularity of the lattice model used and would warrant further investigation. Histogram (Tube width 35, 50000 runs) 1800 1600

No of occurrences

1400 1200 1000 800 600 400 200 0 2

(i)

15

28

41

54

67

80

93

106 119 132 145 158 171 184

Iterations Till Crossing Time

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Histogram (Tube width 40, 100000 runs) 8000 7000

No of occurrences

6000 5000 4000 3000 2000 1000 0 2

14

26

38

50

62

74

86

98

110

122

134

146

158

170

Iterations Till Crossing Time

(j)

Figs 3.60.(a)-(j) First crossing times for simulation of polymer bound at one end in tubes of width 5,7,8,9,10,15,20,30,35&40 respectively. Data is taken from a large number of runs of each simulation from its starting configuration. The x axis represents the number of iterations till first crossing occurred and the y axis the number of runs which terminated at that particular number. The number of runs performed of each particular simulation is noted at the top of reach figure. Histogram (Tube width 40, 100000 runs) Var1 = 100000*1*expon(x, 0.069) 8000 7000

No of occurrences

6000 5000 4000 3000 2000 1000 0 2

14

26

38

50

62

74

86

98

110

122

134

146

158

170

Iterations Till Crossing Time

Fig 3.61.

3.60 (a)-(j) Represent the histograms of first crossing times of the end node of the polymer versus the number of occurrences of crossing at that particular time. The distributions in each case could not be fit with any standard curves Though in 3.60b and 3.61an exponential curve was over laid on the graph 63 | P a g e

and would possibly fit quite well if the occurrences did not drop off rapidly at a close distance to the origin. Nonetheless there is a statistical distribution of crossing time occurrences with a similar shape along the x axis for the different tube widths 3.60a-j. In each case there is a very sharp rise in density from the origin along +x which quickly becomes a rounded peak, the max point, and then heads downwards starts curving to be parallel to the x axis as it approaches zero. The main point of difference between the distributions is that the scales on the x axis vary by many orders of magnitude. 3.60a diminishes to a value around 10% of its max point at about 1.5x10^6 iterations (value on x axis), whereas for a tube of width 7 the value drops to a comparable point at about 10^4 iterations. By a tube width of 40 this has reduced to where the occurrences diminish to about 10% of their peak value at an x axis value of 50. This suggests a strong correlation between crossing time and tube width. Two metrics were selected, first crossing time and average crossing time from which to hopefully discern a predictable relationship.

Average Crossing Time Vs. Tube Width

Scatterplot (Spreadsheet2 2v*10c) 3500

3000

2500

Var2

2000

1500

1000

500

0 0

5

10

15

20

25

30

35

40

45

Var1

Fig 3.62. Plot of Average Time Taken For End Node To Cross Tube Mid Point From Starting Conformation.The x axis represents tube width and y axis average number of iterations till cross.

Figure 3.62 starting at the extreme end of the x axis has a negligibly small value and until value of about 15 there is only a minimal increase with each successive reduction in tube width. Beyond width 15 there is a trend towards larger and larger increases in average crossing time, with the value at tube width 5 (not plotted due to computer problems) reaching 8x10^4. It appears the tube width has a minimal effect on the average crossing until it is below a certain value then the time becomes increasingly larger with each reduction in width thereafter. 64 | P a g e

The software was then altered again to determine the earliest crossing time over a number of runs. The program would run the following cycle: 1. 2. 3. 4. 5.

Set earliest crossing time to arbitrarily large value Run the simulation Terminate if crossing occurs and set earliest crossing time variable to new value Terminate if reaches earliest previous crossing time before crossing occurs Reset all variables except earliest crossing time and repeat from 2

First Crossing Time Vs. Tube Width 2000 1800 1600

First Crossing Time

1400 1200 1000 800 600 400 200 0 -200 0

5

10

15

20

25

30

35

40

45

Tube Width

Fig 3.63. Plot of First Time of Crossing of End Node To Cross Tube Mid Point From Starting Conformation. The x axis represents tube width and y axis average number of iterations from simulation start till crossing of the horizontal midpoint of the tube.

Figure 3.64 shows the first crossing time on the y axis vs. Tube width on the x axis. The values were negligibly small for x values from 40 to 10, sitting at around 2-3. From x axis values of 10 down to 6 values raised marginally to 4-5. The value was somewhat greater at tube width 4 with a first crossing time of ~650 and a much larger value at 5 of ~1800. Due to the increasingly constrained conformation space around the bound origin with decreasing tube width it seems somewhat anomalous that the value at 5 is higher than at width 4. Reviewing the simulation logs reveals that some of the simulations ran for millions of cycles and over 6 hours on a higher lowest crossing time before jumping down to their lowest value. It seems likely that for each case there are certain progressions of the polymer which are highly unlikely but will cause a very low crossing time and that the performed length of simulation time might not have been large enough to attain accurate results. At the least the plot obtained in fig3.62 provides a relationship which is somewhat more clearly defined given the current set of results.

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Chapter 4: Conclusion As persistence length in the lattice model was set to equal one unit lattice cell length, hopefully these results will provide some useful relationships with which can be converted into real world scale using the ratio between the tube width and persistence length as the measure in determining relative sizes of components. Using the relationship described in fig3.62 it should hopefully be possible to control the frequency of the walk cycle to ensure the time spent in the contraction state was significantly smaller than the average first crossing time, ensuring the inchworm propagate in one intended direction for a reasonable amount of time. The values and end node density distributions will also hopefully provide some rough notion of the ideal binding site density and radius, or at least assist in guiding future study.

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Chapter 5: Proposals For Future Study This thesis will hopefully facilitate further investigation into the proposed inchworm motor. The following avenues might be explored. Binding Conditions These values for first, and average crossing time could be recorded and analysed with the binding condition included in the simulation to capture the free end of the polymer. I expect the probabilities of traversal would be hugely diminished with an effectively sized and positioned binding site as the end node would in the vast majority of cases be caught by the binding site and held before having enough time to have any significant chance of flipping over. 3 Dimensional Case Adding an extra dimension to the simulation for a full 3D model would greatly increase the conformation space and no doubt greatly reduce the first and average crossing times for each respective tube width. It would also vastly reduce the chance of the end node finding the binding site in a given time, though this could be remedied somewhat with a model which could bind anywhere along the chain and reptate to the end point. Further study could include investigation of the crossing times in a full 3D model and the difference in probability of binding before crossing in 3D vs. 2D. This would give a good basis for understanding the timescales at which the inchworm could be expected to propagate in a given direction along the x axis. Problems Caused By Increased Conformation Space In 3D And Possible Solution Current fabrication techniques will necessitate the nanotube being rectangular along its cross section. In this case the width in one dimension could likely be machined down to a width comparable to the persistence length of the polymer with zero salinity content. The other dimension being possibly significantly larger. This would seemingly have an adverse affect on the ability to control and minimize the first crossing and average crossing times, as the larger dimension would greatly increase the conformation space. For example, a tube of width ~4 times the persistence length provides very effective suppression of crossing time in the 2 dimensional model and likely a significant amount of suppression in the square 3D tube model. If the relationship between dimensional width and crossing time from fig3.62 is taken as a loose approximation for the behaviour in 3 dimensions, even doubling the width of one dimension would cause a dramatic decrease in crossing times. It is apparent however that due to shielding the persistence length of the polymer is directly related to ionic strength of the surrounding medium and thus theoretically externally controllable. The ionic strength at the release point in time of each walk cycle might be adjusted so as to provide only a partial relaxation of the persistence length, constricting the conformation space and keeping crossing times within desired levels. This would further ensure unidirectional motion and might have other uses as far as fine tuning the system. This might be simulated in future in either a lattice simulation or in a full dynamical model. Adjusted Persistence Length And Its Effect On Inchworm Velocity This increase in persistence length would be somewhat analogous to a reduced nanotube width, and likely cause the most densely visited area of the end node to be further along the tube, away from the tethered origin. This would also likely cause the most effective point to place the capture radius to be 67 | P a g e

further away from the origin and thus the distance traversed by the inchworm each walk cycle. A further area of research could thus be the relationship and trade off between controlling the conformation space to avoid flipping of propagation direction and the distance travelled per walk cycle / speed of propagation for a given length of polymer. The interrelated variables affecting propagation which might be examined in a future simulation thus include: • •

• •

Polymer absolute length – affects statistical distribution of end to end length and thus optimal capture point and propagation speed Persistence length upon release of end node in walk cycle – affects conformation space and likelihood of polymer crossing tube and reversing velocity. Also affects distribution of end to end length and thus propagation speed Thickness of third dimension of tube – affects conformation space and thus flip rate Capture radius – affects optimal speed of walk cycle

But these are all for some future bright researcher to enquire into, this concludes the thesis!

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Chapter 6: References [1] C. Bustamante, Y. R. Chemla, N. R. Forde, D. Izhaky (2004). "Mechanical processes in biology," Annual Review of Biochemistry, 73: 705-748. PMID 15189157

[2] Walter Reisner, Jason P. Beech, Niels B. Larsen (2007). “Nanoconfinement-Enhanced Conformational Response of Single DNA Molecules To Changes in Ionic Environment” Physical Review Letters, PRL, 058302

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