Escape Rates of Externally Confined Polymers

A polymer escaping from a confining external potential represents a generic description of long macromolecules crossing an energy barrier. This type of barrier crossing problems are typical in nano- and microscale polymeric systems, where the polymers are escaping from entropic traps by thermal fluctuations. These systems have possible bioengineering applications, where they can be for example used in sorting polymers. In this thesis, polymer escape from one- and two-dimensional external potentials was studied theoretically and computationally. In a two-dimensional asymmetric external potential, the escape rate of a polymer was solved using Path Integral Hyperdynamics (PIHD) simulations and Kramers' theory using effective potentials for different lengths of polymers. We found that Kramers' theory predicts the escape rate of PIHD simulations qualitatively but the prediction agrees quantitatively only for shorter chains. We also determined that a one-dimensional reaction coordinate is not sufficient to describe the dynamics of the longer polymer chains. In a one-dimensional symmetric double-well external potential, the escape rate was solved using Langevin dynamics simulations, Brownian dynamics simulations, harmonic transition state theory (HTST) with dynamical corrections (DC), Langer's theory, and Forward flux sampling (FFS). FFS and HTST with DC both predict the rate by Langevin and Brownian dynamics simulations quantitatively within a factor of two. We also introduced a new method for computing dynamical corrections using forward flux sampling type of algorithm and compared computational efficiency of the different methodsÍ líftækni er áhugi á kerfum þar sem stórum sameindum er haldið á takmörkuðu svæði með ytra mætti þar til þær sleppa út við það að yfirstíga fríorkuhól fyrir tilstilli varmafræðilegrar örvunar. Dæmi um slík kerfi eru nanó eða míkróskala entrópíugildrur sem notaðar eru til að aðgreina fjölliður. Í þessari ritgerð er lýst kennilegum rannsóknum og reikningum á sleppihraða fjölliða í tví- og þrívíðum kerfum. Sleppihraði fjölliða af ýmsum lengdum var reiknaður fyrir ósamhverft mættisfall með því að nota feriltegur háhreyfijöfnu (PIHD) og aðferð Kramers. Niðurstöður reikninganna voru í mjög góðu samræmi við beina reikninga fyrir stuttar fjölliður en ekki nema í grófu samræmi fyrir langar fjölliður. Þetta sýnir að einvíð hvarfstika nægir ekki til að lýsa færslu langra fjölliða. Sleppihraðinn fyrir samhverft mættisfall með tveimur orkubrunnum var reiknaður með ýmsum aðferðum, svo sem Langevin hreyfijöfnu, Brown hreyfijöfnu, virkjunarástandskenningu innan kjörsveifilsnálgunar (HTST) með leiðréttingu frá tímaferlum (DC) og áframflæði reikningum (FFS). FFS og HTST/DC aðferðirnar gefa báðar mat á sleppihraðanum í góðu samræmi við Langevin og Brown hreyfijöfnur. Ný aðferð til að reikna DC sem nýtir eiginleika FFS var sett fram og reikniþörfin borin saman við aðrar aðferðir.This work has been supported by the FiDiPro programme of the Academy of Finland and travel during this work by the Education Network in Condensed Matter and Materials Physics. Computational resources have been provided by Aalto Science-IT project and CSC - IT Center for Science. The MSP group is part of the Centre of Excellence in Computational Nanoscience (COMP) funded by the Academy of Finland

Driven translocation of a semi-flexible polymer through a nanopore

We study the driven translocation of a semi-flexible polymer through a nanopore by means of a modified version of the iso-flux tension propagation theory (IFTP), and extensive molecular dynamics (MD) simulations. We show that in contrast to fully flexible chains, for semi-flexible polymers with a finite persistence length $\tilde{\ell}_p$ the {\it trans} side friction must be explicitly taken into account to properly describe the translocation process. In addition, the scaling of the end-to-end distance $R_N$ as a function of the chain length $N$ must be known. To this end, we first derive a semi-analytic scaling form for $R_N$, which reproduces the limits of a rod, an ideal chain, and an excluded volume chain in the appropriate limits. We then quantitatively characterize the nature of the {\it trans} side friction based on MD simulations of semi-flexible chains. Augmented with these two factors, the modified IFTP theory shows that there are three main regimes for the scaling of the average translocation time $\tau \propto N^{\alpha}$. In the stiff chain (rod) limit $N/\tilde{\ell}_p \ll 1$, {$\alpha = 2$}, which continuously crosses over in the regime $1 < N/\tilde{\ell}_p < 4$ towards the ideal chain behavior with {$\alpha = 3/2$}, which is reached in the regime $N/\tilde{\ell}_p \sim 10^2$. Finally, in the limit $N/\tilde{\ell}_p \gg 10^6$ the translocation exponent approaches its symptotic value $1+\nu$, where $\nu$ is the Flory exponent. Our results are in good agreement with available simulations and experimental data

Polymer escape from a confining potential

The rate of escape of polymers from a two-dimensionally confining potential well has been evaluated using self-avoiding as well as ideal chain representations of varying length, up to 80 beads. Long timescale Langevin trajectories were calculated using the path integral hyperdynamics method to evaluate the escape rate. A minimum is found in the rate for self-avoiding polymers of intermediate length while the escape rate decreases monotonically with polymer length for ideal polymers. The increase in the rate for long, self-avoiding polymers is ascribed to crowding in the potential well which reduces the free energy escape barrier. An effective potential curve obtained using the centroid as an independent variable was evaluated by thermodynamic averaging and Kramers rate theory then applied to estimate the escape rate. While the qualitative features are well reproduced by this approach, it significantly overestimates the rate, especially for the longer polymers. The reason for this is illustrated by constructing a two-dimensional effective energy surface using the radius of gyration as well as the centroid as controlled variables. This shows that the description of a transition state dividing surface using only the centroid fails to confine the system to the region corresponding to the free energy barrier and this problem becomes more pronounced the longer the polymer is. A proper definition of a transition state for polymer escape needs to take into account the shape as well as the location of the polymer.Peer reviewe

Escape from a One Dimensional Potential Well

Communication: Full dimensional quantum rate coefficients and kinetic isotope effects from ring polymer molecular dynamics for a seven-atom reaction OH + CH4 → CH3 + H2O J. Chem. Phys. 138, 221103 [http://d

Polymeerin pako-ongelman numeeriset simulaatiot

Polymer escape is an event occurring in numerous processes in the nature. Modern DNA sequencing methods as well as some novel medical treatments and drugs are based on this phenomenon. In this Master's Thesis polymer escape from a metastable external potential well by thermal activation was studied. Problem setting corresponded to the Kramers' problem generalised for polymers. Dynamics was obtained by the Langevin equation and the polymer was modelled with the bead-spring model having monomers connected by finite extension nonlinear elastic (FENE) springs and Lennard-Jones (LJ) potential giving excluded volume interactions. To obtain the crossing rate of the escape, the path integral hyperdynamics (PIHD) method was used to enhance the direct simulations of the crossing events. The external potential with perpendicular confinement was introduced and its effects on crossing rate were compared with the original potential. The crossing rate was also calculated by sampling the energy landscapes of the escape event and applying the Kramers' formula and the transition state theory (TST) to these energy landscapes. For the new external potential, a set of PIHD biases was designed and tested. An increase in the crossing rate for longer polymers was found in the new potential. The polymer was also found out to favour escaping in a stretched conformation thus having a strong analogy to the quantum tunnelling of one particle. The Kramers' formula and TST predicted qualitative behaviour of the crossing rate correctly but their quantitative values differed.Polymeerin pako-ongelma esiintyy luonnossa monissa biologisissa prosesseissa. Muun muassa monet modernit DNA:n sekvenssointimenetelmät, lääketieteelliset hoidot sekä lääkkeet hyödyntävät tätä ilmiötä. Tässä diplomityössä tutkittiin termisen aktivaation aiheuttamaa polymeerin pakenemista metastabiilista ulkoisesta potentiaalikuopasta. Ongelman asettelu vastaa kuuluisaa Kramersin ongelmaa yleistettynä polymeereille. Dynamiikka saadaan Langevinin yhtälöstä ja polymeeriä kuvataan helmi-jousi-mallilla, jossa polymeetin monomeerejä yhdistävät epälineaariset FENE-jouset sekä Lennard-Jones potentiaalista saatava äärellinen tilavuus. Pakotaajuus selvitettiin ilmiön numeerisilla simulaatioilla, joita nopeutetaan polkuintegraalihyperdynamiikalla (PIHD). Ylimääräisen kaarevuuden sisältävää uutta ulkoista potentiaalia tutkittiin ja polymeerin käyttäytymistä siinä verrattiin vanhaan potentiaaliin. Pakotaajuus laskettiin myös mallintamalla numeerisesti pakenemisen aikaisia energiapintoja ja soveltamalla Kramersin teoriaa sekä transitiotilateoriaa (TST) näihin pintoihin. Uudelle potentiaalille suunniteltiin neljä ajavaa PIHD-voimaa ja näitä voimia testattiin. Tulosten perusteella pakotaajuus kasvaa pidemmillä polymeereillä uudessa potentiaalissa selkeästi. Polymeeri myös näyttää suosivan venynyttä tilaa paetessaan potentiaalikuopasta. Tämä ilmiö vihjaa analogiasta yhden hiukkasen kvanttitunnelointiongelmaan. Kramersin teorian ja TST:n ennusteet pakotaajuuksille käyttäytyvät kvalitatiivisesti oikein, mutta lukuarvot poikkeavat

Polymeerien pakonopeus rajoittavasta ulkoisesta potentiaalista

A polymer escaping from a confining external potential represents a generic description of long macromolecules crossing an energy barrier. This type of barrier crossing problems are typical in nano- and microscale polymeric systems, where the polymers are escaping from entropic traps by thermal fluctuations. These systems have possible bioengineering applications, where they can be for example used in sorting polymers. In this thesis, polymer escape from one- and two-dimensional external potentials was studied theoretically and computationally.  In a two-dimensional asymmetric external potential, the escape rate of a polymer was solved using Path Integral Hyperdynamics (PIHD) simulations and Kramers' theory using effective potentials for different lengths of polymers. We found that Kramers' theory predicts the escape rate of PIHD simulations qualitatively but the prediction agrees quantitatively only for shorter chains. We also determined that a one-dimensional reaction coordinate is not sufficient to describe the dynamics of the longer polymer chains.  In a one-dimensional symmetric double-well external potential, the escape rate was solved using Langevin dynamics simulations, Brownian dynamics simulations, harmonic transition state theory (HTST) with dynamical corrections (DC), Langer's theory, and Forward flux sampling (FFS). FFS and HTST with DC both predict the rate by Langevin and Brownian dynamics simulations quantitatively within a factor of two. We also introduced a new method for computing dynamical corrections using forward flux sampling type of algorithm and compared computational efficiency of the different methods.Polymeerin pako-ongelma ulkoisesta rajoittavasta potentiaalista on yleinen kuvaus makromolekyylien energiavallin ylityksestä. Tämän tyyppiset energiavallin ylitykset ovat tyypillisiä mikro- ja nanoskaalan polymeerisysteemeistä, joissa polymeeri pakenee termisten fluktuaatioden ansiosta entroppisesta energiakuopasta. Tällaisilla systeemeillä on sovelluskohteita bioinsinööritieteissä, esimerkiksi polymeerien lajittelussa niiden pituuden perusteella. Tässä väitöskirjassa on numeerisesti tutkittu polymeerien pako-ongelmaa yksi- ja kaksiulotteisessa systeemissä.  Kaksiulotteisessa epäsymmetrisessa ulkoisessa potentiaalissa pakonopeus on selvitetty polkuintegraalihyperdynamiikkasimulaatioilla ja Kramersin teorialla käyttäen efektiivisiä potentiaaleja eri mittaisille polymeereille. Kramersin teoria ennustaa pakonopeuden kvalitatiivisesti oikein, mutta ennuste on kvantitatiivisesti tarkka vain lyhyille polymeeriketjuille. Työssä on myös todettu, että yksidimensioinen reaktiokoordinaatti ei ole riittävän tarkka kuvaamaan pakoprosessia pidemmille ketjuille.  Yksiulotteisessa, symmetrisessä ja kaksoiskaivon muotoisessa ulkoisessa potentiaalissa pakonopeus laskettiin käyttämällä Langevinin ja Brownin dynamiikka-simulaatioita, sekä harmonista transitiotilateoriaa dynaamisilla korjauksilla, Langerin teoriaa ja "forward flux sampling"-menetelmää. "Forward flux sampling"-menetelmä sekä harmoninen transitiotilateoria dynaamisilla korjauksilla ennustavat kvantitatiivisen tarkasti Langevinin ja Brownin dynamiikalla lasketun pakonopeuden. Dynaamisille korjauksille esitettiin myös uudentyyppiinen "forward flux sampling" menetelmään perustuva algoritmi sekä verrattiin eri menetelmien laskennallista tehokkuutta

Efficient dynamical correction of the transition state theory rate estimate for a flat energy barrier

The recrossing correction to the transition state theory estimate of a thermal rate can be difficult to calculate when the energy barrier is flat. This problem arises, for example, in polymer escape if the polymer is long enough to stretch between the initial and final state energy wells while the polymer beads undergo diffusive motion back and forth over the barrier. We present an efficient method for evaluating the correction factor by constructing a sequence of hyperplanes starting at the transition state and calculating the probability that the system advances from one hyperplane to another towards the product. This is analogous to what is done in forward flux sampling except that there the hyperplane sequence starts at the initial state. The method is applied to the escape of polymers with up to 64 beads from a potential well. For high temperature, the results are compared with direct Langevin dynamics simulations as well as forward flux sampling and excellent agreement between the three rate estimates is found. The use of a sequence of hyperplanes in the evaluation of the recrossing correction speeds up the calculation by an order of magnitude as compared with the traditional approach. As the temperature is lowered, the direct Langevin dynamics simulations as well as the forward flux simulations become computationally too demanding, while the harmonic transition state theory estimate corrected for recrossings can be calculated without significant increase in the computational effort.Peer reviewe