Description of DNA molecular motion for nanotechnology applications

Progress in Materials Science 74 (2015) 308–331

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Progress in Materials Science journal homepage: www.elsevier.com/locate/pmatsci

Description of DNA molecular motion for nanotechnology applications Firas Awaja a,b,⇑, Edgar A. Wakelin b, Jamie Sage b,c, Abdulmalik Altaee d a

Center for Materials and Microsystems, PAM-SE, Fondazione Bruno Kessler, Trento, Italy School of Physics, University of Sydney, Sydney, NSW 2006, Australia University College London, University of London, London, UK d School of Mechanical Engineering, University of Sydney, NSW 2006, Australia b c

a r t i c l e

i n f o

Article history: Received 24 March 2015 Accepted 25 March 2015 Available online 28 May 2015 Keywords: DNA Molecular motion Diffusion Molecular dynamics

a b s t r a c t One of the most important quests in modern science is the ability to mimic DNA replication, this process provides opportunities to vastly improve current technology from data storage to fighting disease. How molecular processes govern DNA signalling is a key question to be answered for this endeavour to be achieved. One of the greatest difficulties with DNA manipulation has been to physically manipulate individual DNA molecules. In vivo there are a large variety of proteins present which are finely controlled by other proteins and signalling molecules to ensure DNA will coil when required and uncoil for replication when stimulated. Understanding the dynamics of motion of particles and molecules that participate in DNA processes is key to understanding and potentially mimicking these processes. In this review, we introduce the current knowledge with respect to molecular dynamics of motion and draw reference to DNA natural and artificial processes. We also discuss the motion mechanisms and how they might explain the efficiency of DNA signalling and replication processes. Ó 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author at: Center for Materials and Microsystems, PAM-SE, Fondazione Bruno Kessler, Trento, Italy. E-mail address: [email protected] (F. Awaja). http://dx.doi.org/10.1016/j.pmatsci.2015.03.001 0079-6425/Ó 2015 Elsevier Ltd. All rights reserved.

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Contents 1. 2. 3. 4. 5.

6.

7. 8. 9. 10.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Brownian motion and DNA nanotechnology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal or Brownian ratchet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Energy fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The introduction of a bias to Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Energy bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Vibration bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Friction bias: related to the collision theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diffusion models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Diffusion coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. Einstein’s theory for Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4. Solvent collisions (viscosity). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Internal dynamics of DNA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DNA migration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Error corrections mechanism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

309 311 312 312 314 314 315 316 316 316 318 318 319 323 324 326 327 327

1. Introduction DNA nanotechnology is the process of producing both simple and complex 2D and 3D structures out of strands of DNA for the purpose of biological testing and signal processing [1–3]. Two dimensional arrays through crystallographic techniques, molecular electronics and DNA signal computations are currently being developed, allowing for a new era in biological manipulation [4–7]. In living systems, DNA contains information for the entire genome of an organism, when a cell divides, each daughter cell carries an exact copy of these genes and the correct amount of genetic material to prevent defects being introduced. Without such a method of DNA replication, life would have been unable to evolve further than single-celled organisms, and would be unable to reproduce without unacceptable DNA mutations. DNA is a double helix of two single strands of a nucleotide monomer bound together by hydrogen bonding [8–10] forming a double-stranded polymer (Fig. 1). The two strands are antiparallel due to

Fig. 1. The structure of DNA.

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Fig. 2. Schematic of DNA replication process.

the structure of the nucleotide monomers that make up the DNA. Nucleotides are deoxy-ribose sugars attached to a phosphate, forming the ‘backbone’ of the DNA strand, and either one or two nitrogen– carbon rings, known as pyrimidines or purines [11,12] form the nitrogenous bases. The process of DNA replication is required to produce two identical double-stranded helices to ensure genetic conservation along cell lineage [13]. For DNA replication to occur, the two strands which form a double-helix must be unwound and separated to expose the nitrogenous bases of the nucleotides as shown in Fig. 2. A DNA helicase enzyme begins to separate the two strands by breaking the hydrogen bonds joining them. The helical nature of DNA means that to break bonds between the strands causes it to unwind and creates an overabundance of coiling in one direction, known as positive supercoiling. This greatly reduces the effective surface area and accessibility of the DNA strands. In order for DNA polymerase to replicate the DNA, the supercoils must be removed such that the enzyme can physically attach and move along the DNA strands. To accomplish this, a topoisomerase enzyme introduces a cut into both strands, then moves both strands through the created gaps to release the supercoils before re-joining them. This process is highly localised allowing for individual coil rotations to be released, providing specific areas for binding. If the full length of DNA were unwound, it would stretch too far to be contained within a cell so the lengths of DNA outside the working region are kept as supercoils to greatly reduce the length and contain the DNA within the nucleus of the cell [14–16]. The separated strands are complementary to each other due to the nature of purine–pyrimidine base-pairing [17]. An adenine nucleotide base can only hydrogen bond with a thymine base and the same is true for cytosine and guanine. Single nucleotides bind to the start of the available bases in the open DNA strands to form an RNA primer through the activity of a primase enzyme [18]. The primers form the start of the complementary DNA strand and their presence enables a DNA polymerase enzyme to extend the chains in a 50 to 30 direction due to the energy requirements of forming a sugar– phosphate bond [19,20]. Each single DNA strand needs to be replicated to produce two daughter DNA molecules. As polymerases are only able to extend a DNA chain in the 50 to 30 direction, a leading strand and a lagging strand are produced [19,21]. The leading strand can be synthesised in a continuous direction from 50 to 30 , the lagging strand however, requires the polymerase to synthesis smaller DNA fragments. These short DNA fragments only present on the lagging strand are called Okazaki fragments and are subsequently joined together with a DNA ligase enzyme [20,21] to produce two identical daughter strands. This unidirectionality of DNA polymerases was first identified in 1966 [22]. One of the challenges with regards to DNA manipulation has been to physically manipulate individual DNA molecules. In vivo there are many proteins present which are very finely controlled by other proteins and signalling molecules to ensure DNA coils up when required and uncoils for replication when stimulated. For example, by the action of cyclins, proteins which follow a cyclical pattern of expression that signal to the DNA to ensure the replicative cycle is followed correctly [23]. To artificially produce DNA nanostructures, the DNA molecules must be rationally designed to ensure the single-stranded DNA will base-pair with the complementary strands exactly as required. Several problems arise when utilising a self-assembly method. The most significant of these arises from the DNA base-pairing scheme, as erroneous base-pairing can occur at relatively low temperatures. Base mis-pairing forms the basis of the megaprimer approach [24,25] used to produce specific point mutations in DNA sequences.

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There are many limiting steps in DNA replication that have proven to be difficult to reproduce in vitro. For example, the polymerase chain reaction (PCR) [26] represented a highly significant step forward in DNA technology, however, the length of DNA that can be cloned with this technique is limited. In vivo, after a helicase enzyme has separated two DNA strands, the single strand of DNA could fold in on itself producing hairpin loops but is prevented from doing this by local proteins. These proteins are not present in PCR and consequently, to prevent unwanted coiling, the most efficient and accurate reactions are carried out with relatively short lengths of DNA. Additionally, there are no known signalling mechanisms between the DNA and those proteins that facilitate DNA processes even if those proteins were present in a PCR system. RNA primers present in PCR are designed to bind to regions flanking the specific DNA strand to be amplified. These RNA primers have the same function as those in DNA replication performed in vivo but are much longer and require much greater care during design to prevent primers dimerising, aberrant binding, and incorrect replication which all significantly reduce the efficiency of the reaction. If DNA-binding proteins were then included, the amount of reagents and control needed over the reaction to ensure it could be carried out smoothly would inhibit the DNA yield to an even greater extent and give more opportunity for mistakes to arise. High temperatures used in PCR further limit the precision of base-pairing, but the speed at which the reaction is carried out is generally considered to offset the limitations on precision [27]. Therefore, when manufacturing DNA nanostructures, the DNA molecules often need directed assembly to ensure the strands bond in the desired fashion [28]. Directed assembly of DNA nanostructures often requires the use of optical tweezers and other physical manipulation to begin the self-assembly process before assembly can be self-sustained by entropy through complementary base-pairing. More complex DNA nanostructures require more time for the self-assembly mechanism to produce the correct result as any aberrant base-pairs produced will propagate as significant errors. The internal dynamics of DNA play a fundamental role in many biological phenomena and in particular the process of denaturation, that is, the unzipping of the double stranded DNA. Energy fluctuations surrounding the DNA due to its vibrational motion, acts as a signalling process for the helicase to be ratcheted towards it, in a random process to perform the process of unzipping the DNA. In this paper we discuss the science of internal molecular dynamics as well as the effects of Brownian motion, to describe their function as a signalling source for enzymes that play a major role in DNA replication processes, in addition to their ability to produce environments capable to generating complex structures to perform intricate tasks.

2. Brownian motion and DNA nanotechnology The study of Brownian motion is of great importance when considering the future of nanotechnology and in particular DNA technology. There are different interpretations of the role of Brownian motion in describing the molecular/protein interaction with DNA. The Einstein and Smoluchowski model predicts the movement of Brownian particles using a diffusion mechanism and allows for macroscale uses. The Langevin model predicts the movement of particles by utilising Newtonian mechanics and a ‘complimentary force’. In these models, it has been shown both numerically and theoretically that the viscous effects present in Brownian motion are essential for DNA–RNA transcription [29]. Zdravkovic´ et al. [29], considered DNA strands as Brownian particles where the surrounding RNA polymerase (RNAP) enzyme acts as the solvent, this assumption is supported by experimentation where strands of DNA have been transported over a silicon chip displaying Brownian motion, acting as a Brownian ratchet [30]. Brownian motion plays a further role once RNAP binds to the DNA strand, as RNAP moves along the strand transcribing the DNA, it must use energy to produce motion. If RNAP is considered the Brownian particle, the energy required for movement can be obtained from its surroundings. The movement has been shown to closely follow the model of a Brownian ratchet where the energy difference required is in the order of the thermal energy present (10–20 kT) [31]. Changing the focus of the Brownian particle from a strand of DNA to proteins has been shown to be reasonable as proteins have been observed and manipulated as Brownian particles using electrical

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impulses [32]. This description is greatly simplified as it assumes the fluid surrounding a DNA strand is a single particle, however the modelling proposed by Zdravkovic´ and co-researchers [29] confirms that it is not only thermal fluctuations and collisions between Brownian particles that govern the motion, but that the solvent, and its viscosity play a crucial part in the long lasting interactions between DNA and RNA. The equilibrium temperature and thermal fluctuations have been found to significantly affect the movement of Brownian particles throughout a solvent by exciting vibrational and rotational modes within molecules. Within DNA nanotechnology, these excitations can explain the changes in the size of the region of unwound DNA during transcription [33]. 3. Thermal or Brownian ratchet The concept of a thermal ratchet, or Brownian ratchet, is another approach used to study the denaturation of DNA. The concept is based on the view that DNA zipping/unzipping is a biased random process activated by thermal energy. In a Brownian ratchet, a particle representing a biological motor diffuses between equally spaced barriers. However, the ratchet mechanism has the effect of introducing a positive drift to the dynamics of the particle in order to gain information about the asymptotic velocity of the motor, passage times, diffusion parameters, genetic transcription and drug intercalation. One of the interesting utilisations of the idea of a Brownian ratchet, is a Brownian motor based mechanism in the process of unzipping DNA [34]. Based on statistical mechanics and the Boltzmann distribution, in which the internal energy, on a micro level, is represented by a collection of microstates, DNA may be represented in a similar manner. The folded DNA is modelled as having a single state and the unfolded DNA as ‘g’ microstates with nonzero probability. The probability of any state can be calculated based on statistical mechanics. Consequently the zipping/unzipping of DNA can be thought of as a biased probability process due to the fact that the unfolded DNA has ‘g’ microstates compared with a single microstate for folded DNA. The free energy required to activate the various microstates available is supplied by the enzyme helicase. In this model, the helicase generates free energy that is required to unzip the DNA by catalysing ATP-to-ADP hydrolysis. When the helicase encounters a junction in the DNA it transfers some of the free energy to the barrier to unzip it based on a nonzero probability for the ‘g’ microstates of the folded strand. After thermally unzipping the barrier, the helicase moves forward to prevent folding the unzipped barrier. This process may continue until the whole DNA helix is unzipped. 4. Energy fluctuations The change in energy as a result of a Brownian molecule colliding with a solvent particle or the medium has been shown to abide by a probability density function [35,36]:

hDEm i ¼ E  hE0 i

ð1Þ 0

where DEm is the gain in energy of the medium particle, and E the Brownian particle’s energy post-collision, essentially showing conservation of energy. The use of this function supports the assumption that moving particles will continue to collide and exchange energy until they reach thermal equilibrium (see Eqs. (2) and (3) in which T is temperature and C is heat capacity), however, when thermal fluctuations are considered on a nanoscale, nano-regions of particles not at thermal equilibrium can be produced continually. As a result these stochastic collisions will seemingly continue indefinitely (Eq. (4), N is the number of particles)

DE 2 ¼ T 2 C

Classical mechanics

ð2Þ

pffiffiffi DE ¼ T C Classical mechanics

ð3Þ

DE 1  pffiffiffiffi hEi N

ð4Þ

Nanosystems

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The interactions between Brownian particles can be described by elastic collisions [37] where the specific dynamics of the statistically random (stochastic) process is governed by the thermal fluctuations of the solvent [37]. Thermal fluctuations are the random changes in kinetic energy of particles from the thermal equilibrium of the system. The size of the thermal fluctuations are dependent on the temperature [38], where the probability of a particle with a given kinetic energy is governed by the Boltzmann distribution. In the case of DNA nanotechnology, Brownian motion is non-relativistic, and as a result abides by classical mechanics with respect to momentum and kinetic energy. If a collision takes place between a particle of velocity Vi mass Mi and another of velocity Vj, mass Mj the more general case of inelastic conditions are considered, see Eqs. (5) and (6). The resulting velocities post collision can be modelled as follows [39], where Vj0 and Vi0 are the velocities before the collision, a is the restitution coefficient and r is the vector between the two centres of mass (Fig. 3).

v i ¼ v 0i 

mj ^ r ^ ð1 þ aÞ½ðv 0i  v 0j Þr mi þ mj

ð5Þ

v j ¼ v 0j 

mi ^ r ^ ð1 þ aÞ½ðv 0i  v 0j Þr mi þ mj

ð6Þ

From these equations the collisions between particles may be predicted, however due to the vast number of collisions every second, it has been computationally unreasonable to solve any useful system. For example, if a DNA molecule in an ideal gas is considered where the collision rate is taken as the inverse of the mean free path divided by the root mean squared velocity (time between collisions) [40], and the DNA strand is assumed to be a 10 nm diameter sphere whose mass is 33,750 Da, the number of collisions is in the order of 1021 s1. This consideration is for one DNA molecule, useful analysis of Brownian motion requires not only multi molecule models, but in conditions far more dense, such as cellular fluid, increasing the frequency of collisions further. As a result, the tendency so far has been to develop models to deal with macroscopic properties rather than microscopic, however, with the development of high powered computers, it is now possible to model small systems using Monte Carlo Simulations [41]. It is clear from these examples that thermal or Brownian ratchets hold great potential for DNA manipulation through the use of thermally available energy with a positive bias. Some other techniques have also been used to successfully create ratchets such as utilising dielectric potential materials [42], optical tweezers [43] and electrocapillary forces [44,45].

vj vi

v’i v‘j Fig. 3. Collision between two particles.

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5. The introduction of a bias to Brownian motion Biased motion is a general rule that governs Brownian motion and is one of the basic principles in physics that a system not in thermal equilibrium tends towards equilibrium. This can involve a movement in space such that the system never reaches thermal equilibrium in order to maintain movement of the particles. The energy required to achieve this can come from chemical reactions, fluctuating temperature, vibration or by periodic forcing of a Brownian particle. Some examples of biases for Brownian motion in various systems are introduced below. 5.1. Energy bias Energy bias is used when a system is not in thermal equilibrium to invoke directional movement according to:

hDEm i – E  hE0 i þ A

ð7Þ

where DEm is the gain in energy of the medium particle, E the pre-collision energy, E0 the Brownian particle’s energy post-collision, and A is energy bias. One example in biology is the helicase protein which generates the free energy required to unzip DNA by catalysing Adenosine-50 -triphosphate (ATP) to ADP via hydrolysis. Protein motion along microtubule and microfilament polymers within cells is powered by ATP, this ‘power source’ of the cell, which provides energy in the form of chemical energy contained within a tri-phosphate bond can be released when required, and is mediated by specific motor proteins, dynein and actin, each allowing protein motion in one direction only Eqs. (5) and (6). Another example of energy bias is the electrophoreses technique that uses an electric field to cause motion of dispersed particles relative to a fluid, Fig. 4. The migration of DNA using Brownian dynamics is carried out by electrophoresis [46], in which DNA is introduced into a gel, which serves as a random diffuser, and is driven by potential energy that is externally applied across an electrophoresis chamber [47–50]. The larger molecules move slower than the smaller molecules. However standard gel electrophoresis is incapable of separating molecules larger than 40,000 bonds [51], which play a major role in genome projects. In order to achieve a higher separation rate a microfluidic approach is considered, such approach is based on a Brownian ratchet, illustrated in Fig. 5. Many microfludic structures that take advantage of Brownian ratchets, allowing unidirectional motion, have been studied. The idea is based on moving particles through narrow sized obstacles tilted with respect to the flow [32,52–55]. The advantage of Brownian ratchets over other techniques is its ability to sort molecules according to their diffusion coefficients, however the separation of large molecules are slow because diffusion by nature is a slow process. A great improvement in the separation rate of DNA was reported by tilting the electrophoretic flow relative to the vertical axis of the array [52]. The concept of a thermal ratchet is another approach to study how DNA may operate in the presence of substantial thermal motion [56–65] or Brownian ratchet. The concept is based on the observation that a combination of non equilibrium and asymmetry leads to transport. The Brownian motor Force F

Electric field

Fig. 4. Electrophoreses inducing a force F on a dispersed particle.

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Fig. 5. Brownian ratchet (a) on, (b) off.

concept is often used to model biological molecular motors to get some insight into genetic transcription and drug intercalation. One of the interesting utilizations of the Brownian ratchet is a Brownian motor based mechanism of the process of unzipping dsDNA. The unidirectional translocation of the helicase enzyme (acting as a Brownian ratchet here) over the single strand of a dsDNA is stopped by a base pair junction. The frequently occurring thermal fluctuations capable of unzipping the junction will provide the required energy for the helicase to step forward securing the unzipped site and preventing the DNA from zipping up again. The ratio of the stepping rate was given as [34]:

Kþ ¼ eDG =K B T K

ð8Þ

where (K+), (K) are the rate of stepping towards the junction and the rate of stepping away from the junction, respectively. DG is a positive free energy change resulting from the catalyzation of Adenosine-50 -triphosphate (ATP) to adenosine diphosphate (ADP) and coupled to helicase translocation. The drift velocity at which helicase can unzip dsDNA can be given as:

0 h¼b

ðKDGTÞ

@e

ðKDGTÞ

e 1  1a Kþ

B

B

1 A

ð9Þ

where (b) is the length of a base pair, DG is the energy required to unzip one base pair and (a) is the zipping rate. For one base pair, the value of DG can be given as 3kBT compared to DG  16kB T, giving the helicase the required energy to move forward (see Fig. 6). 5.2. Vibration bias Low frequency DNA vibrations have been observed for some time in the order of 10–40 cm1 [66]. Photons of this energy are too low to interact with covalent or ionic bonds, indicating that long-range low energy DNA bonds are being excited. Absorption stimulated by such low energy photons is not typically seen in other biological molecules and as such is speculated to play an important role in biological function [67]. Models describing DNA motion when stimulated by low energy radiation indicate that the molecule undergoes a form of ‘Breathing’ [68] where the base pairs vibrate along their hydrogen bonds and the amplitude of vibration depends on the position within the DNA strand. This phenomenon,

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Fig. 6. Unidirectional motion along the DNA molecule.

where certain base pairs will vibrate differently depending on their position is believed to be key to efficiency of drug and protein intercalation. 5.3. Friction bias: related to the collision theory Conformational changes in motor proteins induced by nucleotide binding or hydrolysis leads to asymmetric internal velocity fluctuations. Coupled with protein friction, those fluctuations result in unidirectional motion along the DNA molecule. 6. Diffusion models Diffusion plays a major role in the ability of proteins to reach a target faster. This process is known as facilitated diffusion and it affects all aspect of DNA dynamics. In this section the dynamics of diffusion is thoroughly discussed. 6.1. Diffusion coefficient The diffusion coefficient of a material describes the rate at which molecules diffuse through materials, and is dependent on the molecule size, the material viscosity and the temperature at which diffusion is occurring. The mathematics underlying diffusion behaviour and the diffusion coefficient plays a pivotal role in the diffusion mechanism itself. Modelling particle diffusion is primarily based on two assumptions, the first being the conservation of the number of particles

@nðx; tÞ @ Cðx; tÞ ¼ @t @x

ð10Þ

where Cðx; tÞ is the particle flux through a point x and nðx; tÞ is the particle density. The second assumption is Fick’s law; that is a relationship between the particle flux Cðx; tÞ and the density nðx; tÞ via a constant which is directly related to the material in question,

Cðx; tÞ ¼ D

@nðx; tÞ @x

where D is the diffusion coefficient [69,70].

ð11Þ

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Fick’s first law is often presented in this form [70–72],

J ¼ D

@/ @x

ð12Þ

where J is the diffusion flux (mol/(m2 s)), D is the diffusion coefficient (m2/s), / represents the concentration per volume (mol/m3) and x is position. Fick’s first law is directly related to the Stokes–Einstein relation, which describes the diffusion coefficient in terms of the viscosity and spherical radius of the molecules which are present in the diffusion process (Eq. (13)).

D¼

KBT 6pgr

ð13Þ

where K B is Boltzmann’s coefficient, T is the temperature in Kelvin, g is the viscosity, and r is the spherical radius. However, Fick’s first law only describes steady state diffusion, whereas most systems undergoing some form of diffusion process do so in a non steady state, or are in fact composite systems. In the latter case, the diffusion coefficients can often be complex functions dependent on many factors including temperature, concentration and molecular weight [73]. Fick’s second law describes diffusion for a non-steady state as follows

@/ @2/ ¼D 2 @t @x

ð14Þ

Eq. (15) was used to calculate the diffusion coefficients before being approximated to the Arrhenius equation [74]. For this approximation D0 is the factor associated with the maximum possible diffusion of the system, E is the activation energy for the reaction where R is the ideal gas law constant and T the absolute temperature. E

D ¼ D0 eRT

ð15Þ

This allowed Font et al. [74] to calculate the apparent activation energies, as the internal diffusion of elastomers is considered to be Fickian in nature. Systems that follow Fickian diffusion have a relaxation time that is related to the diffusive acceleration of molecules and hence is very small. For systems where the relaxation times are larger, for example, diffusion through solid polymers, the Fickian model for diffusion becomes superseded by a hyperbolic differential equation such as:

Nþ/

@N @p ¼ D @t @x

for a stationary solid polymer, where

ð16Þ @N @t

is the change in mass flux with respect to time, D is the dif-

fusion coefficient, / is the mass flux relaxation time and @p is the change in pressure with respect to the @x spatial co-ordinate x axis [75]. It was found that the free-volume theory of transport developed by Vrentas and Duda [76,77] (Eq. (17)) gave a better description than the empirical Arrhenius equation (15). This work was focused on understanding the role of penetrant size and configuration on PMMA and PVA polymers with over 30 solvents tested and diffusion coefficients determined using capillary column inverse gas chromatography (CCIGC), with diffusion described as: cV  f=K 12 2

D1 ¼ D01 eK 22 þTT g2

ð17Þ

where T is the absolute temperature, and f is the ratio between the holes required size to allow a solvent molecule to move (V 1 ) to the required size of the hole which will allow the movement of a polymer jumping unit ðV 2 Þ. (K 22  T g2 ) is dependent on the polymer properties alone while D01 and cV 2 f=K 12 are dependent on the polymer–solvent system. Flexibility and compactness of the material have a major influence on diffusive behaviour [76,77]. The results from penetrant size and configuration indicate that free volume theory itself needs to be re-examined.

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An investigation using poly-isobutylene (PIB) and methyl ethyl ketone (MEK) showed that the diffusion coefficients measured by CCIGC were very similar to those calculated by Vrentas–Duda free volume theory [78]. 6.2. Collisions A typical Brownian particle will undergo 1021 collisions per second [79], which occur predominantly with the surrounding medium rather than other Brownian particles. Previous work on Brownian motion [80] focused on developing collisions based models to study the movement of particles of a dimension approximately 106 m in diameter (for example pollen grains) that exhibit observable Brownian motion. It is logical, therefore, to investigate molecular collisions and dynamics when considering Brownian motion. The change in energy as a result of a Brownian molecule colliding with a solvent particle or the medium has been shown to abide by the same probability density function mentioned earlier in Eq. (1) [81]. The moving particles will continue to collide and exchange energy until they reach thermal equilibrium [82], however when thermal fluctuations are considered on a nanoscale, fluctuations can continually produce nano-regions of particles not at thermal equilibrium. As a result, these stochastic collisions will seemingly continue indefinitely. The result of these collisions is probabilistic in nature and has a range of outcomes [83] whose behaviour is used to model protein and other biological motors [83]. The movement of protein motors however differs from particle diffusion, where a preferential directionality of the motor is predefined. The energy required to produce the action is provided by the stochastic conversion of ATP to ADP [84], the time step between actions can vary due to the stochastic nature of the energy conversion, and is dependent on ATP concentration, motor load and temperature [85]. 6.3. Einstein’s theory for Brownian motion Einstein’s famous analysis of Brownian motion in 1905 was made using the diffusion equation from a model of random molecular motion instead of from the continuity equation and Fick’s Law [86]. Through many observations, Einstein’s explanation for Brownian motion demonstrates that the overall visible motion exactly matches what you would expect if the particles were atoms or molecules (Fig. 7). According to atomic theory, Einstein discovered that there would have to be an observable movement of suspended microscopic particles. Due to the large number of collisions during Brownian motion, classical mechanics is not able to calculate how far a particle travels during each collision. As such Einstein took into account the collective motion of particles. On the basis of kinetic theory, the probability (P) of a particle at point x

Fig. 7. Illustration of Einstein concept of collective Brownian motion.

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at time t, q(x, t) in a medium with known coefficient of diffusion (D) can be calculated according to this initial assumption. When considering the collective motion of Brownian particles and the probability density of Brownian particles at point x at time t, q(x, t), combined with the Diffusion Equation the following equations can be derived:

@q @2q ¼D 2 @t @x P ¼ qðx; tÞ ¼

ð18Þ 1 1=2

ð4pDtÞ

ex

2 =4Dt

ð19Þ

In this formula D is equal to one-half the average of the square of the displacement in the x-direction as expressed below:

x2 ¼ 2Dt

ð20Þ

The Diffusion Equation allowed for calculations of the mean square displacement in terms of the time elapsed and the diffusivity. Einstein then argued that the displacement of a Brownian particle is proportional to that square root of, and not total elapsed time. 6.4. Solvent collisions (viscosity) Within Brownian motion, the particles constituting the solvent and the Brownian particles can change depending on the focus of the study. In the assumptions made in Einstein’s Theory, the numbers of collisions between the particle and solvent have been high, and symmetrical. This implies that the Brownian particles must be much larger than the solvent molecules, and that any molecules with a small physical size and mass compared to the particle of interest can be considered part of the solvent. DNA motion is strongly influenced by solvent viscosity [87–90], where the viscosity causes demodulation of the DNA internal harmonics modes. The harmonic modes determine the shape, vibration and movement of the strand in a solvent, and allow for different rates of bonding and reactivity of the functional groups. The damping effect of the solvent on the DNA is not constant but rather selective, resulting in an optimum functionality of the DNA [29]. Viscosity comes from both the solvent and DNA binding proteins interacting with the DNA, it has been argued that these molecules cause high and low viscosities respectively [91]. At high viscosity the damping effect of the solvent ensures infrequent transitions and long lasting interactions between a DNA strand and the surrounding RNA nucleotide. At low viscosity the DNA behaves in a global harmonic way, where the strand moves as a single body. At a critical viscosity, intermediate between the high and low viscosity, the DNA is more mobile than the two extremes, Figs. 8a and 8b. Smoluchowski’s diffusion model is a combination of both the microscopic predictions and macroscopic results of Brownian motion, linking the two without violating the second law of

Fig. 8a. Brownian particles in a solvent.

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Fig. 8b. Brownian particles in a solvent t = 1.

thermodynamics through the reversibility of motion [92]. It predicts an observable diffusion relation as a result of a large number of unobservable collisions between the Brownian and solvent particles. These collisions provide both the driving and retarding force in the diffusion relation. The Einstein model for Brownian motion on the other hand, does not consider collisions between Brownian particles and the solvent, rather the solvent is viewed as a continuous medium, where the movement depends only on the ‘viscosity of the liquid and the size of the suspended particles’ [86]. Both models here arrive at the same quantitative result of predicting the mean square displacement [92] where the Smoluchowski model is a factor of 64/27 greater than the Einstein model. However it has been shown by Langevin that the predictions of the Smoluchowski model can match exactly the Einstein model under certain assumptions [92]. These two models approach the problem from very different directions, the Einstein model is sufficient when observing or predicting large macroscopic changes in the positions of Brownian Particles such that the medium does not participate in any collisions explicitly. The Smoluchowski model on the other hand investigates the collisions between Brownian particles and the medium through viscosity and mass interactions allowing for greater predictions of movement on the microscopic scale. Both models arrive at the same conclusions on macroscopic movement, which also match experiment, indicating many of the assumptions made are accurate and must be considered when developing further theories to describe Brownian motion. The collisions between particles should be investigated in addition to studying the macroscopic properties of the system when using the Smoluchowski model. If a system containing many low mass solvent molecules in which many high mass Brownian particles exist there will be three types of collisions; solvent–solvent collisions, solvent–Brownian collisions and Brownian–Brownian collisions, each resulting in different exchanges of energy. To explore biological nanotechnology, for the purpose of manipulating DNA and other biological molecules we are mainly interested in the random movement of large Brownian particles in a quasi homogenous and uniform solvent. The solvent–solvent collisions should not affect the Brownian particle motion unless there is a large disturbance in the dispersion or advection capacity of the solvent. This ideal condition rarely occurs however, due to the polarity, non-spherical shape and other solvent factors. The solvent–solvent interactions can and do perform a large role in the movement of Brownian particles which is not limited to a purely viscous nature [93] such as in DNA–RNA transcription [29]. A vast number of collisions occur between Brownian (mass M) and solvent particles (mass m) each second (1021 s1). Each collision imparts a small amount of momentum (DPMm) in a spherically symmetrical way, such that all the collisions appear to cancel each other. However, these collisions, although imparting a small amount of momentum, if summed over a large number, can provide an observable effect. These collisions act as a retarding or viscous force, lengthening the time required for uniform dispersion of the Brownian particles throughout the solvent, as predicted by the

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Langevin model [94]. In this model, Langevin describes the motion of large Brownian particles suspended in a viscous fluid and develops a formula to describe the motion of the particles using Newtonian physics [95], see Eq. (21) below. The last term in the equation below, coined the ‘complementary force’ is the force that counteracts the effects of viscosity and allows the Brownian particles to remain in motion. The role of viscosity in nanoparticle dynamics can be very important such as the example presented in the introduction. 2

m¼

d x 2

d t

¼ 6pla

dx þx dt

ð21Þ

If we imagine Brownian particles as spheres in a solvent, the Einstein diffusion constant and mobility constants can be described as follows [96]:

D ¼ kB Tb

ð22Þ

where:

b ¼ ð6plaÞ1

where b is for translational movement

ð23Þ

This result shows that although coming from highly varied origins, there is great similarity between the three models. They show that a diffusion relation can model the movement of Brownian particles in a solution where the viscosity of the solvent is predicted to have an effected. This has recently been confirmed experimentally through the measurements of the mean displacement squared in a range of fluids with varying viscosity, where the higher the viscosity, the lower the mean displacement squared rate [97,98], a very similar result to the predictions made by the Einstein model and obeys the following relation:

hx2 i ¼

2KT t 3pga

ð24Þ

There are however a few key differences in the way in which the models address the effects and results of the viscosity of the fluid. Langevin describes it as a complimentary force to be found empirically and without explanation, Smoluchowski considered the viscosity and momentum theoretically and as a result the movement is expected, rather than a result of nature. The Einstein model requires secondary relations between the diffusivity constant and properties of the solvent to be solved. Using the Einstein model and these results, it is possible to only consider the solvent as a viscous fluid, which slows the Brownian particles, rather than a sea of atoms providing vast numbers of collisions per second. However, if coagulation or chemical interactions are to occur between the Brownian particles, such as in DNA nanotechnology, then a more sophisticated model must be employed [99]. Brownian–Brownian collisions occur between large particles, in low frequency and in a non-symmetrical fashion. These collisions can result in an observable change in momentum of the Brownian particle, see Eq. (25) and when combined with the viscous and chemical effects of the solvent, can be used to describe the Brownian motion.

DPMM  DP Mm

ð25Þ

This assumption however is only valid if the solvent particles are sufficiently smaller and less massive than the Brownian particles, resulting in the collisions model discussed above. It has been shown that the diffusive coefficient governing Brownian motion is dependent on the effective mass of the collision. When the mass of the solvent particles reach a significant proportion of the Brownian particle (such as in DNA nanotechnology where RNA polymerase is considered the solvent, but not when water is considered the solvent) the diffusion relation is not only dependent on the viscosity of the solvent, but also on the effective mass of the collision [29,98,100]. Considering a large number of Brownian particles such as proteins packed close together in a solvent (with a DNA strand containing the desired binding sites present in the solvent), see Fig. 8a. If the boundary between the Brownian particles and the solvent is investigated, it can be seen that on the solvent side DP Mm is provided per collision whereas on the Brownian particle side DP MM is provided

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per collision, as a result there will be a strong tendency for the Brownian particles to move away from the corner and disperse throughout the solvent. If this phenomenon continues, independent non-coupled Brownian particles will become uniformly distributed throughout the solvent so as to minimise the number of Brownian–Brownian collisions, and will appear to abide by a diffusion model. The Brownian–Brownian collisions do not have to be elastic and one dimensional, they may impart a range of energies to other Brownian particles resulting in a variety of translational, rotational and vibrational momentum which explains the local movement of each Brownian particle. If lone DNA bases are considered (which are not independent, but coupled), it would be expected that if given enough time, the bases would ‘pair off’ and distribute about the solvent. There is the possibility of bond hybridisation between these pairs and water, which has been shown to exist in a small but significant amounts [101]. The movement of Brownian particles throughout a medium however, is not as simple as spheres drifting through a Newtonian fluid where the only effect is that of retardation. A recent study has shown that movement of micro and nano-particles within a crowded environment such as intracellular fluid is highly constrained and anisotropic as well as being dependent on the elastic and mechanical properties of the particle [102]. This indicates that the movement of proteins through intracellular fluid may not be random but directionalised allowing for greater targeted binding, Fig. 9.

Fig. 9. Protein movements in a viscous fluid towards DNA.

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A uniform distribution should occur if there is no positive binding force. In the case of the diagram shown in Fig. 9, the DNA strand present in the solvent will provide selective hydrogen-binding sites and will result in proteins within the solvent binding. As the protein–DNA structure is formed, the hydrogen binding sites from the DNA groups will be occupied, and as a result, the remaining Brownian particles will distribute uniformly throughout the solvent. This does not suggest that the particles stop moving, rather the density of the Brownian particles per unit volume will remain constant. This concept is supported by results presented by Keyser [103] where it has been shown that the movement of particles within a water solvent subject to solvent and self interactions (i.e. hydrophilic and hydrophobic interactions such as in proteins) can significantly change the particles movement throughout a medium through a change in the power spectral density (i.e. non-random movements) and thermal interactions with its surroundings on the nanometer scale. 7. Internal dynamics of DNA The DNA structure consists of flexible and moveable parts. These causing can undergo complex internal movements, such as vibrations of atoms about an equilibrium position, rotation, linear and transverse motion that contribute greatly to the process of denaturation [104]. The internal vibrational motion of the DNA is greatly influenced by the medium surrounding it in addition to other factors such as the interactions with proteins and drugs [105–107]. The solvent or the surrounding medium acts as a large reservoir of energy that can supply or absorb heat without a considerable change in temperature. The temperature of the surrounding medium acts as a source of energy required to activate the dynamic motion of DNA causing it to vibrate enharmonically, where such motion requires a temperature above 220 K [105–107]. In addition to the heat supplied; the surrounding medium also contributes to the dynamic motion of DNA through random collisions. The very nature of this dynamic movement of DNA will cause an energy fluctuation around which will be generated an energy well that will allow ratcheting of molecules such as helicase acting as a signalling source. The dynamics of DNA has been investigated experimentally using neutron spectroscopy, a suitable tool to measure atomic activities because of its ability to measure density fluctuation. Neutron spectroscopy is employed to measure the activation energy, amplitude of vibration and relaxation time of DNA. It has been shown that a sharp increase of the mean-squared atomic displacements is temperature dependent and takes place at about 200–230 K [105,108,109]. A series of mechanical models have been suggested to represent the structure of DNA. The models attempted to describe the internal DNA molecular motion in order to obtain an analytical solution to the process of unzipping DNA. These models predicted the shape and magnitude of the energy required to separate the double strands and explains the nature of the unzipping. Traditional mechanical models [110–116] have assumed that DNA is a one dimensional lattice composed of N base pair units; each nucleotide is represented by a point mass m. These masses are joined along the strand by a spring of constant strength k to represent the stacking interaction between nearest bases on each strand. The two strands on the opposite sides are stabilized by hydrogen bonds, which are represented by an external potential; such as the model shown in Fig. 11. Different parts of the model shown in Fig. 10 can undergo different motions [112,117–123], to simplify the analysis; it was assumed that the rotational motion of the bases provided the main contribution to the unzipping of DNA. Other studies assumed that the transverse motion of the hydrogen

Lennard-Johns potential Point mass Stacking energy Fig. 10. One dimensional lattice representation of DNA.

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Disc of mass Morris potential

Covalent bond Fig. 11. More realistic representation of DNA.

bonds is the main contributor to the opening of the DNA. Further work included both motions, and assumed that the hydrogen bonding was represented by a Lennard-Johns potential. Calculations based on such models have successfully reproduced DNA melting and unzipping data for a variety of DNA sequences [121,124–130]. An example of such work was published by Voulgarakis et al. [130] in which a numerical Monte Carlo approach was used to describe the mechanical unzipping of DNA based on a simple one dimensional model proposed by Peyrard–Bishop–Dauxois [121]. The work was able to quantitatively reproduce a force–temperature phase diagram that was obtained experimentally by Danilowicz et al. [108]. A more realistic model is to represent each nucleotide as a disc of mass m [112], consequently the rotational kinetic energy of the nucleotide must be included (Fig. 11). In this model the hydrogen bonding was represented by more practical barrier; the Morris Barrier. In order to obtain an analytical solution describing the dynamic motion of DNA; two different methods may be considered, Newton’s second law of motion or the energy method. The latter is easier to use in this case since it is a scalar method and only requires the calculation of the kinetics and potential energy of the system without considering the internal forces acting on each particle. To achieve an analytical solution utilising the energy method; a well-known mathematical formulation, the Hamiltonian, is considered. A Hamiltonian that includes the DNA interactions; and describes its different motions may be written as [131]:

H¼

N X Hðyi ; yiþ1 Þ

ð26Þ

i¼1

Hðyi ; yiþ1 Þ ¼

P 2i þ Wðyi ; yiþ1 Þ þ Vðyi Þ 2m

ð27Þ

This is a general energy equation to describe most if not all mechanical models. 8. DNA migration Investigation of the migration of DNA using Brownian dynamics has been carried out using electrophoresis [132], where DNA is introduced to a gel serving as a random diffuser [30], and is driven by a potential externally applied across electrophoresis chamber. The larger molecules move slower than the smaller molecules, however standard gel electrophoresis is incapable of separating molecules larger than 40,000 Da [52], which play a major role in genome projects. In order to achieve a higher separation rate, a microfluidic approach is considered [133] in which microfabricated structures are used with alternating deep regions and shallow constrictions. Entering the constrictions is thus entropically penalized, and the deep regions can be interpreted as entropic traps that retard the motion of small molecules more than the larger ones, hence achieving separation [133].

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Another microfludic structure that takes advantage of Brownian ratchets, allowing unidirectional motion has been constructed [32,53]. It is based on moving particles through narrow sized obstacles tilted with respect to the flow [54,55]. The advantage of Brownian ratchets over other techniques is its ability to sort molecules according to their diffusion coefficients [134], however the separation of large molecules are slow because diffusion by nature is a slow process. A great improvement in the separation rate of DNA was reported by tilting the electrophoretic flow relative to the vertical axis of the array [52]. The Brownian motion of confined DNA has been reported by Nykypanchuk et al. [109] in which DNA molecules were confined to two-dimensional wells where it was observed that the molecules can ‘jump’ between two adjacent cavities. This type of motion is well described by Poisson statistics where the flexibility of the DNA chains allow it to contort and travel through holes connecting each cavity. DNA is a very flexible molecule due to the bonding angles present between nucleotides, and can undergo a spontaneous helix–coil transition. The helix–coil transition occurs largely due to the interaction forces between neighbouring bases and intra-chain interaction bonding but is also dependent on temperature and factors governing DNA flexibility. The flexibility of the DNA is dependent heavily on the salt concentration e.g. at 10 mM of a monovalent salt, the persistent length of DNA is approximately 150 base pairs (50 nm) [135,136] but at 0.1 mM monovalent salt the persistent length is approximately 750 base pairs (350 nm) [136,137]. The persistent length of DNA is defined as the average length of straight-chained DNA before a thermally excited bend of one radian occurs in the chain. The persistence length of DNA has also been observed to depend on the sequence of nucleotides with the sequence causing a difference of up to 15 nm [138]. The importance of Brownian motion of DNA has been known since 1978 [139,140] but the actual motion of free DNA has not been well observed. Despite this, Brownian motion is described sufficiently by mathematics for simulations to be modelled for several key interactions of DNA. These simulations include, but are not limited to the: binding of two different DNA-bound proteins at distant sites which are required to interact with each other; to initiate specific processes [141]; diffusion of DNA molecules tied into knots with optical tweezers along a stretched DNA chain [142]; Brownian motion of DNA anchored at a site with a traceable particle at the unanchored end [143,144], and other notable simulations [145]. Although Brownian motion is not well understood, it has been recognized as vital for almost all cellular processes, being the most feasible process for intracellular translocation of molecules, including most proteins. Many proteins in the cell are transported to specific, localised and specialized regions by motion along charged rod-like polymers known as microtubules [146] or microfilaments [147]. The cell is a very dynamic entity that requires specific transport of molecules, such as proteins in the axons of neuronal cells, where these lack the machinery to undergo protein synthesis. The proteins must be synthesised in the cell body and then transported down the length of the axon. The motion along microtubule and microfilament polymers is powered by ATP. Understandably, this requires the cell to exert very fine control over microtubules to ensure that signalling molecules arrive at their required location. Once a protein has attached to the microtubule the subsequent motion can be described by Brownian motion, which allows the proteins to travel along the microtubule scaffolding [148]. The heavy chain ‘head’ domain of the motor protein is attached to its cargo, the protein the cell needs to transport, and the light chain ‘tail’ domain is attached to the microtubule. Motor proteins then ‘walk’ along the microtubule with their Y-shaped tail and when one of the ‘feet’ dissociates from the microtubule, it walks a step forward through Brownian motion and the energy from ATP [149–151]. A summary of steps that represent the DNA replication mechanism is shown in Fig. 9. These molecular motor proteins behave slightly differently to polymerases, discussed earlier, which need to overcome Brownian motion in order to stay attached to the DNA and perform their designated task. During replication, any errors introduced to the DNA strand must be rectified and the enzyme which removes the erroneous bases does so through Brownian motion without the requirement of ATP for energy [152]. This last point is very important as the removal of bases occurs through a Brownian ratchet mechanism eliminating the need for an energy source (see Fig. 12). Modern efforts to design a Brownian ratchet are focused on producing a bias on the random thermal motion to supply a force. Such Brownian ratchets would need to allow thermally activated motion only in a single direction to produce controlled motion from thermal fluctuations. The ratchets are

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Fig. 12. Summary of DNA replication mechaism.

abundant in cellular systems and utilised by many proteins for molecule translocation [153]. Unidirectional motion can be accomplished in rotary molecular systems [154] by enforcing a strong friction on one direction of rotation. As rotary systems only have motion in two directions, it is possible to place a biased friction on the ‘backwards’ rotation to produce directed motion. Proton pumps in mitochondria, which use reducing power as the energy source for rotating the pump, can move in either direction but a strong bias influences the motion to ensure that the protons are always pumped against their diffusion gradient, where the pump will not turn in the opposite direction allowing the diffusion gradient to decrease. Work has since been done in attempts to produce ‘stepping’ motors which function similarly to the protein motors for transport along microtubules [155] in cellular environments [156]. It has been observed that the unidirectional motor proteins which ‘walk’ along the microtubules have a small probability of stepping in the wrong direction, suggesting that the bias is very strong but not absolute [83,157].

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9. Error corrections mechanism All living organisms require a proofreading mechanism to ensure that DNA chains have been replicated properly. Even a single base error can result in disastrous consequences, for example the genetic condition cystic fibrosis is caused by a single nucleotide error in mRNA, which produces a single amino acid change in its associated protein. There are three mechanisms cells utilise to ensure extremely high accuracy in replication: base selection by the polymerase enzyme; proofreading also performed by the polymerase; and mismatch repair carried out by the mutHLS system, known as an antimutator [158]. As mentioned previously, polymerases can only add nucleotides to the 3-primed end of the single stranded DNA chain due to the energy requirement of the free phosphate group to extend the chain. Most DNA polymerases possess a 30 –50 exonuclease activity that removes incorrectly inserted bases by moving backwards over the DNA chain. The incorrect nucleotide is believed to stall the polymerase enzyme and decrease the probability of forward motion, thereby increasing the probability of the polymerase stepping backwards to correct the mistake. Hence, the Brownian ratchet mechanisms of polymerases’ forward motion are monitored and can be readily changed to maintain high accuracy through a proofreading system [159]. Mismatch repair occurs through a well documented methyl-directed mismatch system [160–162]. The newly synthesised DNA strand does not possess the methylated bases the template strand would have required for some signalling events. This targets DNA correction to the newly synthesised strand in which a section of DNA nucleotides will be removed and then re-added by a DNA polymerase that is recruited to the site. The combination of these three mechanisms allows DNA polymerases to have an error rate of less than 1010.

10. Conclusions DNA nanotechnology is a new and emerging area of science with the potential to affect society in ways ranging from genetic engineering to data storage. Brownian motion has been shown to play an integral role in DNA motion and activity, however the exact way to model and utilise this motion is still not fully defined. The variety of models indicates that there are many layers of complexity to Brownian motion and that a universal general solution has not yet been described. Brownian/thermal ratchets however provide a real example of how the motion can be harnessed for highly efficient and accurate motion on the nanoscale, allowing for increasingly intricate and complex tasks. The dynamic movement of DNA, which is basically vibrations of atoms about an equilibrium position and rotation and transverse molecular motion, results in energy fluctuation in the surrounding medium that allows ratcheting of molecules such as helicase acting as a signalling source. Different modelling techniques were employed to define DNA internal dynamics motion especially towards defining the motion of unzipping. Some studies assumed that the rotational motion of the bases provided the main contribution to the unzipping of DNA. Other studies assumed that the transverse motion of the hydrogen bonds is the main contributor to the opening of the DNA. Further work included both motions, and assumed that the hydrogen bonding was represented by a Lennard-Johns potential. Solvent collision Brownian motion brings together the Einstein and Smoluchowski models where the solvent can provide the medium for diffusion as well as the collision forces to maintain motion, and a viscous or retarding force to prevent excessive movement. The models approach the system from different starting points but arrive at the same conclusion – that Brownian particles tend to diffuse through a solvent and that the solvent, through its viscous nature or individual collisions, has large effect on the motion. Collisions between Brownian particles are easier to model as they occur less frequently and have a larger effect on the macroscopic system. There is now plenty of evidence that Brownian motion, in particular of DNA molecules, is affected by these collisions and that these models are critical in understanding the movement.

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