# Publications

## Articles in Refereed Scientific Journals

**2023**

## Petra Papež, Franci Merzel, Matej Praprotnik

Sub-THz acoustic excitation of protein motion

J. Chem. Phys. 159, 135101, 2023.

**Abstract**

The application of terahertz radiation has been shown to affect both protein structure and cellular function. As the key to such structural changes lies in the dynamic response of a protein, we focus on the susceptibility of the protein’s internal dynamics to mechanical stress induced by acoustic pressure waves. We use the open-boundary molecular dynamics method, which allows us to simulate the protein exposed to acoustic waves. By analyzing the dynamic fluctuations of the protein subunits, we demonstrate that the protein is highly susceptible to acoustic waves with frequencies corresponding to those of the internal protein vibrations. This is confirmed by changes in the compactness of the protein structure. As the amplitude of the pressure wave increases, even larger deviations from average positions and larger changes in protein compactness are observed. Furthermore, performing the mode-projection analysis, we show that the breathing-like character of collective modes is enhanced at frequencies corresponding to those used to excite the protein.

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## Petra Papež, Franci Merzel, Matej Praprotnik

Rotational dynamics of a protein under shear flow studied by the Eckart frame formalism

J. Phys. Chem. B 127, 7231-7243, 2023.

**Abstract**

Proteins are natural polymers that play an essential role in both living organisms and biotechnological applications. During certain bioprocessing steps, they can be exposed to significant mechanical stress induced by, for example, shear flow or sonication, resulting in reduced therapeutic efficacy, aggregation, or even a loss of activity. For this reason, there is a need to understand and determine the susceptibility of the protein activity to the experienced mechanical stress. To acquire this knowledge, it is necessary to study the rotational dynamics of the protein. Commonly, the rotational dynamics of soft molecules is interpreted based on a theoretical analysis performed in an inertial laboratory frame. However, the obtained angular velocity mixes pure rotations and vibrations with angular momentum, consequently lacking a clear dynamical interpretation. On the other hand, the use of the noninertial internal Eckart frame allows the determination of pure angular velocity as it minimizes the coupling between the rotational and vibrational degrees of freedom. In the present work, by conducting open-boundary molecular dynamics simulations and exploiting the Eckart frame formalism, we study the rotational dynamics of a small protein under the shear flow of various strengths. Our results show that the angular velocity increases nonlinearly with increasing shear rate. Furthermore, the protein gains vibrational angular momentum at higher shear rates, which is reflected in the higher angular velocity computed by employing the Eckart frame formalism and confirmed by analysis of the contributions to the total kinetic energy of the biomolecule.

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## Tilen Potisk, Jurij Sablić, Daniel Svenšek, Elena Sanz-de Diego, Francisco J. Teran, Matej Praprotnik

Analyte-driven clustering of bio-conjugated magnetic nanoparticles

Adv. Theory Simul. 6, 2200796, 2023.

**Abstract**

The dynamics of bio-conjugated magnetic nanoparticles suspended in buffer-saline solutions containing target proteins (i.e., analytes) is investigated numerically on a mesoscopic level. To simulate the dispersion of magnetic nanoparticles the dissipative particle dynamics is employed, which allows to study rather large systems, while still retaining important microscopic nanoparticle properties. In addition, the method is coupled to the Landau–Lifshitz–Gilbert equation, describing the dynamics of the magnetic nanocrystals within the macrospin approximation. The binding of multivalent analytes to the recognition ligands of the nanoparticles leads to the formation of clusters of magnetic nanoparticles, which in turn drastically changes the macroscopic magnetic response of the solution. Such colloidal changes are experimentally observable, allowing to explore new approaches to quantify the analyte amount. The ratio of the concentrations between the analytes (biomarkers) and the recognition ligands on the nanoparticles is found to play an important role in the formation and hydrodynamic size of the clusters. The proposed computational framework has great potential to be integrated with experimental efforts to advance the development of nanoparticle-based biosensors for disease diagnostics and other biomedical applications.

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**2022**

## Aljaž Draškovič-Bračun, Tilen Potisk, Matej Praprotnik, Daniel Svenšek

Suspension of discrete microscopic oscillators as a model of an ultrasonic metafluid

Phys. Rev. B 105, 224317, 2022.

**Abstract**

We present a model of ultrasonic metafluids—acoustic metamaterials in the form of suspensions of discrete microscopic oscillators coupled to the embedding fluid. Contrary to a common assumption about metamaterials, and as already established in the field of metafluids, the metafluid concept need not be based on position periodicity or correlation of the suspended micro-oscillators, and in this case not even on ideally designed micro-oscillators. For the speculation that metafluids may one day be produced as solutions of macromolecules, it is essential that the micro-oscillators be allowed to be randomly distributed in the host fluid and generally have irregular (modal) shapes. We formulate the detailed operating principle of such a metafluid model, give explicit formulas for its effective dynamic moduli in terms of the modal structure of the micro-oscillators, and discuss basic practical issues of performance optimization in terms of their mass and size.

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## Petra Papež, Matej Praprotnik

Dissipative Particle Dynamics simulation of ultrasound propagation through liquid water

J. Chem. Theory Comput. 18, 1227-1240, 2022.

**Abstract**

Ultrasound is widely used as a noninvasive method in therapeutic and diagnostic applications. These can be further optimized by computational approaches, as they allow for controlled testing and rational optimization of the ultrasound parameters, such as frequency and amplitude. Usually, continuum numerical methods are used to simulate ultrasound propagating through different tissue types. In contrast, ultrasound simulations using particle description are less common, as the implementation is challenging. In this work, a dissipative particle dynamics model is used to perform ultrasound simulations in liquid water. The effects of frequency and thermostat parameters are studied and discussed. We show that frequency and thermostat parameters affect not only the attenuation but also the computed speed of sound. The present study paves the way for development and optimization of a virtual ultrasound machine for large-scale biomolecular simulations.

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## Pantelis R. Vlachas, Julija Zavadlav, Matej Praprotnik, Petros Koumoutsakos

Accelerated simulations of molecular systems through Learning of Effective Dynamics

J. Chem. Theory Comput. 18, 538-549, 2022.

**Abstract**

Simulations are vital for understanding and predicting the evolution of complex molecular systems. However, despite advances in algorithms and special purpose hardware, accessing the time scales necessary to capture the structural evolution of biomolecules remains a daunting task. In this work, we present a novel framework to advance simulation time scales by up to 3 orders of magnitude by learning the effective dynamics (LED) of molecular systems. LED augments the equation-free methodology by employing a probabilistic mapping between coarse and fine scales using mixture density network (MDN) autoencoders and evolves the non-Markovian latent dynamics using long short-term memory MDNs. We demonstrate the effectiveness of LED in the Müller–Brown potential, the Trp cage protein, and the alanine dipeptide. LED identifies explainable reduced-order representations, i.e., collective variables, and can generate, at any instant, all-atom molecular trajectories consistent with the collective variables. We believe that the proposed framework provides a dramatic increase to simulation capabilities and opens new horizons for the effective modeling of complex molecular systems.

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**2021**

## Ermioni Papadopoulou, Julija Zavadlav, Rudolf Podgornik, Matej Praprotnik, Petros Koumoutsakos

Tuning the dielectric response of water in nanoconfinement through surface wettability

ACS Nano 15, 20311-20318, 2021.

**Abstract**

The tunable polarity of water can be exploited in emerging technologies including catalysis, gas storage, and green chemistry. Recent experimental and theoretical studies have shown that water can be rendered into an effectively apolar solvent under nanoconfinement. We furthermore demonstrate, through molecular simulations, that the static dielectric constant of water can be modified by changing the wettability of the confining material. We find the out-of-plane dielectric response to be highly sensitive to the level of confinement and can be reduced up to 40×, in accordance with experimental data. By altering the surface wettability from superhydrophilic to superhydrophobic, we observe a 36% increase for the out-of-plane and a 31% decrease for the in-plane dielectric constants. Our findings demonstrate the feasibility of tunable water polarity, a phenomenon with great potential for scientific and technological impact.

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## Núria López, Luigi Del Debbio, Marc Baaden, Matej Praprotnik, Laura Grigori, Catarina Simões, Serge Bogaerts, Florian Berberich, Thomas Lippert, Janne Ignatius, Philippe Lavocat, Oriol Pineda, Maria Grazia Giuffreda, Sergi Girona, Dieter Kranzlmüller, Michael M. Resch, Gabriella Scipione, Thomas Schulthess

Lessons learned from urgent computing in Europe: Tackling the COVID-19 pandemic

Proc. Natl. Acad. Sci. USA 118, e2024891118, 2021.

**Abstract**

PRACE (Partnership for Advanced Computing in Europe), an international not-for-profit association that brings together the five largest European supercomputing centers and involves 26 European countries, has allocated more than half a billion core hours to computer simulations to fight the COVID-19 pandemic. Alongside experiments, these simulations are a pillar of research to assess the risks of different scenarios and investigate mitigation strategies. While the world deals with the subsequent waves of the pandemic, we present a reflection on the use of urgent supercomputing for global societal challenges and crisis management.

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## Robinson Cortes-Huerto, Matej Praprotnik, Kurt Kremer, Luigi Delle Site

From adaptive resolution to molecular dynamics of open systems

Eur. Phys. J. B 94, 189, 2021.

**Abstract**

We provide an overview of the Adaptive Resolution Simulation method (AdResS) based on discussing its basic principles and presenting its current numerical and theoretical developments. Examples of applications to systems of interest to soft matter, chemical physics, and condensed matter illustrate the method’s advantages and limitations in its practical use and thus settle the challenge for further future numerical and theoretical developments.

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**2020**

## Stephan Thaler, Matej Praprotnik, Julija Zavadlav

Back-mapping augmented adaptive resolution simulation

J. Chem. Phys. 153, 164118, 2020.

**Abstract**

Concurrent multiscale techniques such as Adaptive Resolution Scheme (AdResS) can offer ample computational advantages over conventional atomistic (AT) molecular dynamics simulations. However, they typically rely on aphysical hybrid regions to maintain numerical stability when high-resolution degrees of freedom (DOFs) are randomly re-inserted at the resolution interface. We propose an Energy Minimized AT (DOF) Insertion (EMATI) method that uses an informed rather than random AT DOF insertion to tackle the root cause of the issue, i.e., overlapping AT potentials. EMATI enables us to directly couple AT and coarse-grained resolutions without any modifications of the interaction potentials. We exemplify AdResS-EMATI in a system of liquid butane and show that it yields improved structural and thermodynamic properties at the interface compared to competing AdResS approaches. Furthermore, our approach extends the applicability of the AdResS without a hybrid region to systems for which force capping is inadequate.

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## Luigi Delle Site, Matej Praprotnik, John B. Bell, Rupert Klein

Particle–continuum coupling and its scaling regimes: Theory and applications

Adv. Theory Simul. 3, 1900232, 2020.

**Abstract**

This work is motivated by the goal of designing simulation software for technical devices that, at their functional core, rely on atomistic‐scale processes embedded in a larger‐scale fluid environment. The core of the problem is the conceptual and technical approach for coupling particle and continuum representations of a fluid. The state of the art for key aspects including physical modeling, mathematical formalization, computational implementation, and applications, is discussed and organized in a consistent picture across the relevant physical regimes.

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**2019**

## Aleksandar Popadić, Daniel Svenšek, Rudolf Podgornik, Matej Praprotnik

Density–nematic coupling in isotropic linear polymers: Acoustic and osmotic birefringence

Adv. Theory Simul. 2, 1900019, 2019.

**Abstract**

Linear polymers and other connected “line liquids” exhibit a geometrical coupling between density and equilibrium orientational order on the macroscopic level that gives rise to a Meyer‐de Gennes vectorial conservation law for polar orientational order, or its amended version for apolar nematic order when described as “recovered” polar order. They generally exhibit fluctuations of orientational order, starting with its lowest moment, the polar order, which in the isotropic phase is geometrically decoupled from density. As a contrast, quadrupolar (nematic) orientational fluctuations are inherently coupled to density fluctuations already in the isotropic system and not subject to the existence of an orientational phase transition. To capture this, it takes the tensorial description of the nematic order, leading to a geometrical coupling between density and orientational order in the form of a tensorial conservation law. This coupling implies that a spatial density variation will induce nematic order and thereby an acoustic or osmotic optical birefringence even in isotropic phase. The theory is validated by performing detailed Monte Carlo simulations of isotropic melts and comparing the results with macroscopic predictions. This also exposits a means of determining the macroscopic parameters by microscopic simulations to yield realistic continuum models of specific polymeric materials.

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## Julija Zavadlav, Siewert J. Marrink, Matej Praprotnik

SWINGER: A clustering algorithm for concurrent coupling of atomistic and supramolecular liquids

Interface Focus 9, 20180075, 2019.

**Abstract**

In this contribution, we review recent developments and applications of a dynamic clustering algorithm SWINGER tailored for the multiscale molecular simulations of biomolecular systems. The algorithm on-the-fly redistributes solvent molecules among supramolecular clusters. In particular, we focus on its applications in combination with the adaptive resolution scheme, which concurrently couples atomistic and coarse-grained molecular representations. We showcase the versatility of our multiscale approach on a few applications to biomolecular systems coupling atomistic and supramolecular water models such as the well-established MARTINI and dissipative particle dynamics models and provide an outlook for future work.

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**2018**

## Aleksandar Popadić, Daniel Svenšek, Rudolf Podgornik, Kostas Daoulas, Matej Praprotnik

Splay–density coupling in semiflexible main-chain nematic polymers with hairpins

Soft Matter 14, 5898-5905, 2018.

**Abstract**

A main-chain nematic polymer melt/solution exhibits macroscopic orientational order of main polymer chains, i.e., a preferred (nematic) direction. It has long been known that in such polymeric liquid crystals spatial density/concentration variations and distortions of the nematic direction are coupled, obeying a vectorial continuity constraint whose rigidity increases with chain length. Its vectorial nature precludes the application to flexible chains, where backfolds (hairpins) are present and apolar nematic symmetry is manifest, which has been its puzzling feature from the beginning. We now establish a description of the splay--density coupling in the case of arbitrary backfolding, devising a continuity constraint for the "recovered" polar order of the chain tangents and introducing hairpins as its new type of sources. Performing detailed Monte Carlo simulations of nematic monodomain melts of "soft" worm-like chains with variable length and flexibility, we show via their structure factors that the weakening of the coupling due to the backfolding can be consistently quantified on the macroscopic level.

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## Christian Krekeler, Animesh Agarwal, Christoph Junghans, Matej Praprotnik, Luigi Delle Site

Adaptive resolution molecular dynamics technique: Down to the essential

J. Chem. Phys. 149, 024104, 2018.

**Abstract**

We investigate the role of the thermodynamic (TD) force as an essential and sufficient technical ingredient for an efficient and accurate adaptive resolution algorithm. Such a force applied in the coupling region of an adaptive resolution molecular dynamics setup assures thermodynamic equilibrium between atomistically resolved and coarse-grained regions, allowing the proper exchange of molecules. We numerically prove that indeed for systems as relevant as liquid water and 1,3-dimethylimidazolium chloride ionic liquid, the combined action of the TD force and thermostat allows for computationally efficient and numerically accurate simulations, beyond the current capabilities of adaptive resolution setups, which employ switching functions in the coupling region.

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## Julija Zavadlav, Jurij Sablić, Rudolf Podgornik, Matej Praprotnik

Open-Boundary Molecular Dynamics of a DNA molecule in a hybrid explicit/implicit salt solution

Biophys. J. 114, 2352-2362, 2018.

**Abstract**

The composition and electrolyte concentration of the aqueous bathing environment have important consequences for many biological processes and can profoundly affect the behavior of biomolecules. Nevertheless, because of computational limitations, many molecular simulations of biophysical systems can be performed only at specific ionic conditions: either at nominally zero salt concentration, i.e., including only counterions enforcing the system’s electroneutrality, or at excessive salt concentrations. Here, we introduce an efficient molecular dynamics simulation approach for an atomistic DNA molecule at realistic physiological ionic conditions. The simulations are performed by employing the open-boundary molecular dynamics method that allows for simulation of open systems that can exchange mass and linear momentum with the environment. In our open-boundary molecular dynamics approach, the computational burden is drastically alleviated by embedding the DNA molecule in a mixed explicit/implicit salt-bathing solution. In the explicit domain, the water molecules and ions are both overtly present in the system, whereas in the implicit water domain, only the ions are explicitly present and the water is described as a continuous dielectric medium. Water molecules are inserted and deleted into/from the system in the intermediate buffer domain that acts as a water reservoir to the explicit domain, with both water molecules and ions free to enter or leave the explicit domain. Our approach is general and allows for efficient molecular simulations of biomolecules solvated in bathing salt solutions at any ionic strength condition.

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## Julija Zavadlav, Siewert J. Marrink, Matej Praprotnik

Multiscale simulation of protein hydration using the SWINGER dynamical clustering algorithm

J. Chem. Theory Comput. 14, 1754-1761, 2018.

**Abstract**

To perform computationally efficient concurrent multiscale simulations of biological macromolecules in solution, where the all-atom (AT) models are coupled to supramolecular coarse-grained (SCG) solvent models, previous studies resorted to modified AT water models, such as the bundled-simple point charge (SPC) models, that use semiharmonic springs to restrict the relative movement of water molecules within a cluster. Those models can have a significant impact on the simulated biomolecules and can lead, for example, to a partial unfolding of a protein. In this work, we employ the recently developed alternative approach with a dynamical clustering algorithm, SWINGER, which enables a direct coupling of original unmodified AT and SCG water models. We perform an adaptive resolution molecular dynamics simulation of a Trp-Cage miniprotein in multiscale water, where the standard SPC water model is interfaced with the widely used MARTINI SCG model, and demonstrate that, compared to the corresponding full-blown AT simulations, the structural and dynamic properties of the solvated protein and surrounding solvent are well reproduced by our approach.

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## Rudolf Podgornik, Julija Zavadlav, Matej Praprotnik

Molecular dynamics simulation of high density DNA arrays

Computation 6, 3, 2018.

**Abstract**

Densely packed DNA arrays exhibit hexagonal and orthorhombic local packings, as well as a weakly first order transition between them. While we have some understanding of the interactions between DNA molecules in aqueous ionic solutions, the structural details of its ordered phases and the mechanism governing the respective phase transitions between them remains less well understood. Since at high DNA densities, i.e., small interaxial spacings, one can neither neglect the atomic details of the interacting macromolecular surfaces nor the atomic details of the intervening ionic solution, the atomistic resolution is a sine qua non to properly describe and analyze the interactions between DNA molecules. In fact, in order to properly understand the details of the observed osmotic equation of state, one needs to implement multiple levels of organization, spanning the range from the molecular order of DNA itself, the possible ordering of counterions, and then all the way to the induced molecular ordering of the aqueous solvent, all coupled together by electrostatic, steric, thermal and direct hydrogen-bonding interactions. Multiscale simulations therefore appear as singularly suited to connect the microscopic details of this system with its macroscopic thermodynamic behavior. We review the details of the simulation of dense atomistically resolved DNA arrays with different packing symmetries and the ensuing osmotic equation of state obtained by enclosing a DNA array in a monovalent salt and multivalent (spermidine) counterions within a solvent permeable membrane, mimicking the behavior of DNA arrays subjected to external osmotic stress. By varying the DNA density, the local packing symmetry, and the counterion type, we are able to analyze the osmotic equation of state together with the full structural characterization of the DNA subphase, the counterion distribution and the solvent structural order in terms of its different order parameters and consequently identify the most important contribution to the DNA-DNA interactions at high DNA densities.

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**2017**

## Eduardo R. Cruz-Chu, Ermioni Papadopoulou, Jens H. Walther, Aleksandar Popadić, Gengyun Li, Matej Praprotnik, Petros Koumoutsakos

On phonons and water flow enhancement in carbon nanotubes

Nat. Nanotechnol. 12, 1106-1108, 2017.

## Julija Zavadlav, Matej Praprotnik

Adaptive resolution simulations coupling atomistic water to dissipative particle dynamics

J. Chem. Phys. 147, 114110, 2017.

**Abstract**

Multiscale methods are the most efficient way to address the interlinked spatiotemporal scales encountered in soft matter and molecular liquids. In the literature reported hybrid approaches span from quantum to atomistic, coarse-grained, and continuum length scales. In this article, we present the hybrid coupling of the molecular dynamics (MD) and dissipative particle dynamics (DPD) methods, bridging the micro- and mesoscopic descriptions. The interfacing is performed within the adaptive resolution scheme (AdResS), which is a linear momentum conserving coupling technique. Our methodology is hence suitable to simulate fluids on the micro/mesoscopic scale, where hydrodynamics plays an important role. The presented approach is showcased for water at ambient conditions. The supramolecular coupling is enabled by a recently developed clustering algorithm SWINGER that assembles, disassembles, and reassembles clusters as needed during the course of the simulation. This allows for a seamless coupling between standard atomistic MD and DPD models. The developed framework can be readily applied to various applications in the fields of materials and life sciences, e.g., simulations of phospholipids and polymer melts, or to study the red blood cells behavior in normal and disease states.

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## Jurij Sablić, Rafael Delgado-Buscalioni, Matej Praprotnik

Application of the Eckart frame to soft matter: Rotation of star polymers under shear flow

Soft Matter 13, 6988-7000, 2017.

**Abstract**

The Eckart co-rotating frame is used to analyze the dynamics of star polymers under shear flow, either in melt or solution and with different types of bonds. This formalism is compared with the standard approach used in many previous studies on polymer dynamics, where an apparent angular velocity ω is obtained from the relation between the tensor of inertia and angular momentum. A common mistake is to interpret ω as the molecular rotation frequency, which is only valid for rigid-body rotation. The Eckart frame, originally formulated to analyze the infrared spectra of small molecules, dissects different kinds of displacements: vibrations without angular momentum, pure rotation, and vibrational angular momentum (leading to a Coriolis cross-term). The Eckart frame co-rotates with the molecule with an angular frequency Ω obtained from the Eckart condition for minimal coupling between rotation and vibration. The standard and Eckart approaches are compared with a straight description of the star's dynamics taken from the time autocorrelation of the monomer positions moving around the molecule's center of mass. This is an underdamped oscillatory signal, which can be described by a rotation frequency ωR and a decorrelation rate Γ. We consistently find that Ω coincides with ωR, which determines the characteristic tank-treading rotation of the star. By contrast, the apparent angular velocity ω < Ω does not discern between pure rotation and molecular vibrations. We believe that the Eckart frame will be useful to unveil the dynamics of semiflexible molecules where rotation and deformations are entangled, including tumbling, tank-treading motions and breathing modes.

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## Julija Zavadlav, Staš Bevc, Matej Praprotnik

Adaptive resolution simulations of biomolecular systems

Eur. Biophys. J. 46, 821-835, 2017.

**Abstract**

In this review article, we discuss and analyze some recently developed hybrid atomistic–mesoscopic solvent models for multiscale biomolecular simulations. We focus on the biomolecular applications of the adaptive resolution scheme (AdResS), which allows solvent molecules to change their resolution back and forth between atomistic and coarse-grained representations according to their positions in the system. First, we discuss coupling of atomistic and coarse-grained models of salt solution using a 1-to-1 molecular mapping—i.e., one coarse-grained bead represents one water molecule—for development of a multiscale salt solution model. In order to make use of coarse-grained molecular models that are compatible with the MARTINI force field, one has to resort to a supramolecular mapping, in particular to a 4-to-1 mapping, where four water molecules are represented with one coarse-grained bead. To this end, bundled atomistic water models are employed, i.e., the relative movement of water molecules that are mapped to the same coarse-grained bead is restricted by employing harmonic springs. Supramolecular coupling has recently also been extended to polarizable coarse-grained water models with explicit charges. Since these coarse-grained models consist of several interaction sites, orientational degrees of freedom of the atomistic and coarse-grained representations are coupled via a harmonic energy penalty term. The latter aligns the dipole moments of both representations. The reviewed multiscale solvent models are ready to be used in biomolecular simulations, as illustrated in a few examples.

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## Julija Zavadlav, Rudolf Podgornik, Matej Praprotnik

Order and interactions in DNA arrays: Multiscale molecular dynamics simulation

Sci. Rep. 7, 4775, 2017.

**Abstract**

While densely packed DNA arrays are known to exhibit hexagonal and orthorhombic local packings, the detailed mechanism governing the associated phase transition remains rather elusive. Furthermore, at high densities the atomistic resolution is paramount to properly account for fine details, encompassing the DNA molecular order, the contingent ordering of counterions and the induced molecular ordering of the bathing solvent, bringing together electrostatic, steric, thermal and direct hydrogen-bonding interactions, resulting in the observed osmotic equation of state. We perform a multiscale simulation of dense DNA arrays by enclosing a set of 16 atomistically resolved DNA molecules within a semi-permeable membrane, allowing the passage of water and salt ions, and thus mimicking the behavior of DNA arrays subjected to external osmotic stress in a bathing solution of monovalent salt and multivalent counterions. By varying the DNA density, local packing symmetry, and counterion type, we obtain osmotic equation of state together with the hexagonal-orthorhombic phase transition, and full structural characterization of the DNA subphase in terms of its positional and angular orientational fluctuations, counterion distributions, and the solvent local dielectric response profile with its order parameters that allow us to identify the hydration force as the primary interaction mechanism at high DNA densities.

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## Luigi Delle Site, Matej Praprotnik

Molecular systems with open boundaries: Theory and simulation

Phys. Rep. 693, 1-56, 2017.

**Abstract**

Typical experimental setups for molecular systems must deal with a certain coupling to the external environment, that is, the system is open and exchanges mass, momentum, and energy with its surroundings. Instead, standard molecular simulations are mostly performed using periodic boundary conditions with a constant number of molecules. In this review, we summarize major development of simulation methodologies, which, contrary to standard techniques, open up the boundaries of a molecular system and allow for exchange of energy and matter with the environment, in and out of equilibrium. In particular, we construct the review around the open boundary simulation approaches based on the Adaptive Resolution Scheme (AdResS), which seamlessly couples different levels of resolution in molecular simulations. Ideas and theoretical concepts used in its development lie at the crossroad of different fields and disciplines and open many different directions for future developments in molecular simulation. We examine progress related to theoretical as well as novel modeling approaches bridging length scales from quantum to the continuum description and report on their application in various molecular systems. The outlook of the review is dedicated to the perspective of how to further incorporate rigorous theoretical approaches such as the Bergmann–Lebowitz and Emch–Sewell models into the molecular simulation algorithms and stimulate further development of open boundary simulation methods and their application.

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## Jurij Sablić, Matej Praprotnik, Rafael Delgado-Buscalioni

Deciphering the dynamics of star molecules in shear flow

Soft Matter 13, 4971-4987, 2017.

**Abstract**

This work analyses the rotation of star polymers under shear flow, in melts, and in good solvent dilute solution. The latter is modeled by single molecule Brownian hydrodynamics, while melts are modeled using non-equilibrium molecular dynamics in closed (periodic) boxes and in open boundaries. A Dissipative Particle Dynamics (DPD) thermostat introduces pairwise monomer friction in melts at will, in directions normal and tangent to the monomer–monomer vectors. Although tangential friction is seldom modeled, we show that it is essential to control hydrodynamic effects in melts. We analyze the different sources of molecular angular momentum in solution and melts and distinguish three dynamic regimes as the shear rate is increased. These dynamic regimes are related with the disruption of the different relaxation mechanisms of the star in equilibrium. Although strong differences are found between harmonic springs and finitely extensible bonds, above a critical shear rate the star molecule has a “breathing” mode with successive elongations and contractions in the flow direction with frequency Ω. The force balance in the flow direction unveils a relation between Ω and the orientation angle. Using literature results for the tumbling of rings and linear chains, either in melt or in solution, we show that the relation is general. A different “tank-treading” dynamics determines the rotation of monomers around the center of mass of the molecule. We show that the tank-treading frequency does not saturate but keeps increasing with the shear rate. This is at odds with previous studies which erroneously calculated the molecular angular frequency, used as a proxy for tank-treading.

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**2016**

## Julija Zavadlav, Siewert J. Marrink, Matej Praprotnik

Adaptive resolution simulation of supramolecular water: The concurrent making, breaking, and remaking of water bundles

J. Chem. Theory Comput. 12, 4138-4145, 2016.

**Abstract**

The adaptive resolution scheme (AdResS) is a multiscale molecular dynamics simulation approach that can concurrently couple atomistic (AT) and coarse-grained (CG) resolution regions, i.e., the molecules can freely adapt their resolution according to their current position in the system. Coupling to supramolecular CG models, where several molecules are represented as a single CG bead, is challenging but it provides higher computational gains and connection to the established MARTINI CG force field. Difficulties that arise from such coupling have been so far bypassed with bundled AT water models, where additional harmonic bonds between oxygen atoms within a given supramolecular water bundle are introduced. While these models simplify the supramolecular coupling, they also cause in certain situations spurious artifacts, such as partial unfolding of biomolecules. In this work, we present a new clustering algorithm SWINGER that can concurrently make, break and remake water bundles and in conjunction with the AdResS permits the use of original AT water models. We apply our approach to simulate a hybrid SPC/MARTINI water system and show that the essential properties of water are correctly reproduced with respect to the standard monoscale simulations. The developed hybrid water model can be used in biomolecular simulations, where a significant speed up can be obtained without compromising the accuracy of the AT water model.

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## Julija Zavadlav, Rudolf Podgornik, Manuel N. Melo, Siewert J. Marrink, Matej Praprotnik

Adaptive resolution simulation of an atomistic DNA molecule in MARTINI salt solution

Eur. Phys. J. Special Topics 225, 1595-1607, 2016.

**Abstract**

We present a dual-resolution model of a deoxyribonucleic acid (DNA) molecule in a bathing solution, where we concurrently couple atomistic bundled water and ions with the coarse-grained MARTINI model of the solvent. We use our fine-grained salt solution model as a solvent in the inner shell surrounding the DNA molecule, whereas the solvent in the outer shell is modeled by the coarse-grained model. The solvent entities can exchange between the two domains and adapt their resolution accordingly. We critically asses the performance of our multiscale model in adaptive resolution simulations of an infinitely long DNA molecule, focusing on the structural characteristics of the solvent around DNA. Our analysis shows that the adaptive resolution scheme does not produce any noticeable artifacts in comparison to a reference system simulated in full detail. The effect of using a bundled-SPC model, required for multiscaling, compared to the standard free SPC model is also evaluated. Our multiscale approach opens the way for large scale applications of DNA and other biomolecules which require a large solvent reservoir to avoid boundary effects.

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## Jurij Sablić, Matej Praprotnik, Rafael Delgado-Buscalioni

Open boundary molecular dynamics of sheared star-polymer melts

Soft Matter 12, 2416-2439, 2016.

**Abstract**

Open boundary molecular dynamics (OBMD) simulations of a sheared star polymer melt under isothermal conditions are performed to study the rheology and molecular structure of the melt under a fixed normal load. Comparison is made with the standard molecular dynamics (MD) in periodic (closed) boxes at a fixed shear rate (using the SLLOD dynamics). The OBMD system exchanges mass and momentum with adjacent reservoirs (buffers) where the external pressure tensor is imposed. Insertion of molecules in the buffers is made feasible by implementing there a low resolution model (blob-molecules with soft effective interactions) and then using the adaptive resolution scheme (AdResS) to connect with the bulk MD. Straining with increasing shear stress induces melt expansion and a significantly different redistribution of pressure compared with the closed case. In the open sample, the shear viscosity is also a bit lowered but more stable against the viscous heating. At a given Weissenberg number, molecular deformations and material properties (recoverable shear strain and normal stress ratio) are found to be similar in both setups. We also study the modelling effect of normal and tangential friction between monomers implemented in a dissipative particle dynamics (DPD) thermostat. Interestingly, the tangential friction substantially enhances the elastic response of the melt due to a reduction of the kinetic stress viscous contribution.

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**2015**

## Julija Zavadlav, Rudolf Podgornik, Matej Praprotnik

Adaptive resolution simulation of a DNA molecule in salt solution

J. Chem. Theory Comput. 11, 5035-5044, 2015.

**Abstract**

We present a multiscale simulation of a DNA molecule in 1 M NaCl salt solution environment, employing the adaptive resolution simulation approach that allows the solvent molecules, i.e., water and ions, to change their resolution from atomistic to coarse-grained and vice versa adaptively on-the-fly. The region of high resolution moves together with the DNA center-of-mass so that the DNA itself is always modeled at high resolution. We show that our multiscale simulations yield a stable DNA–solution system, with statistical properties similar to those produced by the conventional all-atom molecular dynamics simulation. Special attention is given to the collective properties, such as the dielectric constant, as they provide a sensitive quality measure of our multiscale approach.

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## Rafael Delgado-Buscalioni, Jurij Sablić, Matej Praprotnik

Open boundary molecular dynamics

Eur. Phys. J. Special Topics 224, 2331-2349, 2015.

**Abstract**

This contribution analyzes several strategies and combination of methodologies to perform molecular dynamic simulations in open systems. Here, the term open indicates that the total system has boundaries where transfer of mass, momentum and energy can take place. This formalism, which we call Open Boundary Molecular Dynamics (OBMD), can act as interface of different schemes, such as Adaptive Resolution Scheme (AdResS) and Hybrid continuum-particle dynamics to link atomistic, coarse-grained (CG) and continuum (Eulerian) fluid dynamics in the general framework of fluctuating Navier-Stokes equations. The core domain of the simulation box is solved using all-atom descriptions. The CG layer introduced using AdResS is located at the outer part of the open box to make feasible the insertion of large molecules into the system. Communications between the molecular system and the outer world are carried out in the outer layers, called buffers. These coupling preserve momentum and mass conservation laws and can thus be linked with Eulerian hydrodynamic solvers. In its simpler form, OBMD allows, however, to impose a local pressure tensor and a heat flux across the system’s boundaries. For a one component molecular system, the external normal pressure and temperature determine the external chemical potential and thus the independent parameters of a grand-canonical ensemble simulation. Extended ensembles under non-equilibrium stationary states can also be simulated as well as time dependent forcings (e.g. oscillatory rheology). To illustrate the robustness of the combined OBMD-AdResS method, we present simulations of star-polymer melts at equilibrium and in sheared flow.

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## Rafael Delgado-Buscalioni, Jurij Sablić, Matej Praprotnik

Reply to comments by R. Klein on "Open boundary molecular dynamics"

Eur. Phys. J. Special Topics 224, 2511-2513, 2015.

**Abstract**

We agree with prof. Klein [1] that there are some similarities between our method [2] and the one presented in Ref. [3] (see also Ref. [4]). There are, however, several relevant differences we would like to outline.

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## Aleksandar Popadić, Matej Praprotnik, Petros Koumoutsakos, Jens H. Walther

Continuum simulations of water flow past fullerene molecules

Eur. Phys. J. Special Topics 224, 2321-2330, 2015.

**Abstract**

We present continuum simulations of water flow past fullerene molecules. The governing Navier-Stokes equations are complemented with the Navier slip boundary condition with a slip length that is extracted from related molecular dynamics simulations. We find that several quantities of interest as computed by the present model are in good agreement with results from atomistic and atomistic-continuum simulations at a fraction of the cost. We simulate the flow past a single fullerene and an array of fullerenes and demonstrate that such nanoscale flows can be computed efficiently by continuum flow solvers, allowing for investigations into spatiotemporal scales inaccessible to atomistic simulations.

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## Julija Zavadlav, Manuel N. Melo, Siewert J. Marrink, Matej Praprotnik

Adaptive resolution simulation of polarizable supramolecular coarse-grained water models

J. Chem. Phys. 142, 244118, 2015.

**Abstract**

Multiscale simulations methods, such as adaptive resolution scheme, are becoming increasingly popular due to their significant computational advantages with respect to conventional atomistic simulations. For these kind of simulations, it is essential to develop accurate multiscale water models that can be used to solvate biophysical systems of interest. Recently, a 4-to-1 mapping was used to couple the bundled-simple point charge water with the MARTINI model. Here, we extend the supramolecular mapping to coarse-grained models with explicit charges. In particular, the two tested models are the polarizable water and big multiple water models associated with the MARTINI force field. As corresponding coarse-grained representations consist of several interaction sites, we couple orientational degrees of freedom of the atomistic and coarse-grained representations via a harmonic energy penalty term. This additional energy term aligns the dipole moments of both representations. We test this coupling by studying the system under applied static external electric field. We show that our approach leads to the correct reproduction of the relevant structural and dynamical properties.

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## Staš Bevc, Christoph Junghans, Matej Praprotnik

STOCK: Structure mapper and online coarse-graining kit for molecular simulations

J. Comput. Chem. 36, 467-477, 2015.

**Abstract**

We present a web toolkit STructure mapper and Online Coarse-graining Kit for setting up coarse-grained molecular simulations. The kit consists of two tools: structure mapping and Boltzmann inversion tools. The aim of the first tool is to define a molecular mapping from high, for example, all-atom, to low, that is, coarse-grained, resolution. Using a graphical user interface it generates input files, which are compatible with standard coarse-graining packages, for example, Versatile Object-oriented Toolkit for Coarse-graining Applications and DL_CGMAP. Our second tool generates effective potentials for coarse-grained simulations preserving the structural properties, for example, radial distribution functions, of the underlying higher resolution model. The required distribution functions can be provided by any simulation package. Simulations are performed on a local machine and only the distributions are uploaded to the server. The applicability of the toolkit is validated by mapping atomistic pentane and polyalanine molecules to a coarse-grained representation. Effective potentials are derived for systems of TIP3P (transferable intermolecular potential 3 point) water molecules and salt solution. The presented coarse-graining web toolkit is available at http://stock.cmm.ki.si.

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**2014**

## Aleksandar Popadić, Jens H. Walther, Petros Koumoutsakos, Matej Praprotnik

Continuum simulations of water flow in carbon nanotube membranes

New J. Phys. 16, 082001, 2014.

**Abstract**

We propose the use of the Navier–Stokes equations subject to partial-slip boundary conditions to simulate water flows in Carbon NanoTube (CNT) membranes. The finite volume discretizations of the Navier–Stokes equations are combined with slip lengths extracted from molecular dynamics (MD) simulations to predict the pressure losses at the CNT entrance as well as the enhancement of the flow rate in the CNT. The flow quantities calculated from the present hybrid approach are in excellent agreement with pure MD results while they are obtained at a fraction of the computational cost. The method enables simulations of system sizes and times well beyond the present capabilities of MD simulations. Our simulations provide an asymptotic flow rate enhancement and indicate that the pressure losses at the CNT ends can be reduced by reducing their curvature. More importantly, our results suggest that flows at nanoscale channels can be described by continuum solvers with proper boundary conditions that reflect the molecular interactions of the liquid with the walls of the nanochannel.

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## Julija Zavadlav, Manuel N. Melo, Ana V. Cunha, Alex H. de Vries, Siewert J. Marrink, Matej Praprotnik

Adaptive resolution simulation of MARTINI solvents

J. Chem. Theory Comput. 10, 2591-2598, 2014.

**Abstract**

We present adaptive resolution molecular dynamics simulations of aqueous and apolar solvents using coarse-grained molecular models that are compatible with the MARTINI force field. As representatives of both classes of solvents we have chosen liquid water and butane, respectively, at ambient temperature. The solvent molecules change their resolution back and forth between the atomistic and coarse-grained representations according to their positions in the system. The difficulties that arise from coupling to a coarse-grained model with a multimolecule mapping, for example, 4-to-1 mapping in the case of the Simple Point Charge (SPC) and MARTINI water models, could be successfully circumvented by using bundled water models. We demonstrate that the presented multiscale approach faithfully reproduces the structural and dynamical properties computed by reference fully atomistic molecular dynamics simulations. Our approach is general and can be used with any atomistic force field to be linked with the MARTINI force field.

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## Julija Zavadlav, Manuel N. Melo, Siewert J. Marrink, Matej Praprotnik

Adaptive resolution simulation of an atomistic protein in MARTINI water

J. Chem. Phys. 140, 054114, 2014.

**Abstract**

We present an adaptive resolution simulation of protein G in multiscale water. We couple atomistic water around the protein with mesoscopic water, where four water molecules are represented with one coarse-grained bead, farther away. We circumvent the difficulties that arise from coupling to the coarse-grained model via a 4-to-1 molecule coarse-grain mapping by using bundled water models, i.e., we restrict the relative movement of water molecules that are mapped to the same coarse-grained bead employing harmonic springs. The water molecules change their resolution from four molecules to one coarse-grained particle and vice versa adaptively on-the-fly. Having performed 15 ns long molecular dynamics simulations, we observe within our error bars no differences between structural (e.g., root-mean-squared deviation and fluctuations of backbone atoms, radius of gyration, the stability of native contacts and secondary structure, and the solvent accessible surface area) and dynamical properties of the protein in the adaptive resolution approach compared to the fully atomistically solvated model. Our multiscale model is compatible with the widely used MARTINI force field and will therefore significantly enhance the scope of biomolecular simulations.

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**2013**

## Staš Bevc, Christoph Junghans, Kurt Kremer, Matej Praprotnik

Adaptive resolution simulation of salt solutions

New J. Phys. 15, 105007, 2013.

**Abstract**

We present an adaptive resolution simulation of aqueous salt (NaCl) solutions at ambient conditions using the adaptive resolution scheme. Our multiscale approach concurrently couples the atomistic and coarse-grained models of the aqueous NaCl, where water molecules and ions change their resolution while moving from one resolution domain to the other. We employ standard extended simple point charge (SPC/E) and simple point charge (SPC) water models in combination with AMBER and GROMOS force fields for ion interactions in the atomistic domain. Electrostatics in our model are described by the generalized reaction field method. The effective interactions for water–water and water–ion interactions in the coarse-grained model are derived using structure-based coarse-graining approach while the Coulomb interactions between ions are appropriately screened. To ensure an even distribution of water molecules and ions across the simulation box we employ thermodynamic forces. We demonstrate that the equilibrium structural, e.g. radial distribution functions and density distributions of all the species, and dynamical properties are correctly reproduced by our adaptive resolution method. Our multiscale approach, which is general and can be used for any classical non-polarizable force-field and/or types of ions, will significantly speed up biomolecular simulation involving aqueous salt.

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**2012**

## Jens H. Walther, Matej Praprotnik, Evangelos M. Kotsalis, Petros Koumoutsakos

Multiscale simulation of water flow past a C540 fullerene

J. Comput. Phys. 231, 2677-2681, 2012.

**Abstract**

We present a novel, three-dimensional, multiscale algorithm for simulations of water flow past a fullerene. We employ the Schwarz alternating overlapping domain method to couple molecular dynamics (MD) of liquid water around the C540 buckyball with a Lattice–Boltzmann (LB) description for the Navier–Stokes equations. The proposed method links the MD and LB domains using a fully three-dimensional interface and coupling of velocity gradients. The present overlapping domain method implicitly preserves the flux of mass and momentum and bridges flux-based and Schwarz domain decomposition algorithms. We use this method to determine the slip length and hydrodynamic radius for water flow past a buckyball.

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**2011**

## Matej Praprotnik, Simon Poblete, Kurt Kremer

Statistical physics problems in adaptive resolution computer simulations of complex fluids

J. Stat. Phys. 145, 946-966, 2011.

**Abstract**

Simulating complex fluids or in general complex molecular systems requires approaches covering decades of time and length scales. This usually cannot be achieved within one simulation model. Over the years many different methods and models have been developed ranging from rather generic models, representing most efficiently the universal statistical mechanical properties of e.g. polymers, to all atom models and even quantum mechanical treatments. While these allow for scientifically very important studies in their own right, only a combination and close link between models of different levels allows for a truly quantitative description of materials and processes. In the present contribution we discuss an adaptive resolution approach where different levels of detail are treated within one simulation and the molecules are free to diffuse between different regions in space, where the molecules interact with different interaction potentials.

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## Matej Praprotnik, Simon Poblete, Luigi Delle Site, Kurt Kremer

Comment on "Adaptive multiscale molecular dynamics of macromolecular fluids"

Phys. Rev. Lett. 107, 099801, 2011.

**Abstract**

A Comment on the Letter by S. O. Nielsen, P. B. Moore, and B. Ensing, Phys. Rev. Lett. 105, 237802 (2010). The authors of the Letter offer a Reply.

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## Staš Bevc, Janez Konc, Jure Stojan, Milan Hodošček, Matej Penca, Matej Praprotnik, Dušanka Janežič

ENZO: A web tool for derivation and evaluation of kinetic models of enzyme catalyzed reactions

PLoS ONE 6, e22265, 2011.

**Abstract**

We describe a web tool ENZO (Enzyme Kinetics), a graphical interface for building kinetic models of enzyme catalyzed reactions. ENZO automatically generates the corresponding differential equations from a stipulated enzyme reaction scheme. These differential equations are processed by a numerical solver and a regression algorithm which fits the coefficients of differential equations to experimentally observed time course curves. ENZO allows rapid evaluation of rival reaction schemes and can be used for routine tests in enzyme kinetics. It is freely available as a web tool, at http://enzo.cmm.ki.si.

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**2010**

## Simon Poblete, Matej Praprotnik, Kurt Kremer, Luigi Delle Site

Coupling different levels of resolution in molecular simulations

J. Chem. Phys. 132, 114101, 2010.

**Abstract**

Simulation schemes for liquids or strongly fluctuating systems that allow to change the molecular representation in a subvolume of the simulation box while preserving the equilibrium with the surroundings introduce conceptual problems of thermodynamic consistency. In this work we present a general scheme based on thermodynamic arguments which ensures a thermodynamic equilibrium among molecules of different representations. The robustness of the algorithm is tested for two examples, namely, an adaptive resolution simulation, atomistic/coarse grained, for a liquid of tetrahedral molecules, and an adaptive resolution simulation of a binary mixture of tetrahedral molecules and spherical solutes.

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**2009**

## Rafael Delgado-Buscalioni, Kurt Kremer, Matej Praprotnik

Coupling atomistic and continuum hydrodynamics through a mesoscopic model: Application to liquid water

J. Chem. Phys. 131, 244107, 2009.

**Abstract**

We have conducted a triple-scale simulation of liquid water by concurrently coupling atomistic, mesoscopic, and continuum models of the liquid. The presented triple-scale hydrodynamic solver for molecular liquids enables the insertion of large molecules into the atomistic domain through a mesoscopic region. We show that the triple-scale scheme is robust against the details of the mesoscopic model owing to the conservation of linear momentum by the adaptive resolution forces. Our multiscale approach is designed for molecular simulations of open domains with relatively large molecules, either in the grand canonical ensemble or under nonequilibrium conditions.

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## Matej Praprotnik, Silvina Matysiak, Luigi Delle Site, Kurt Kremer, Cecilia Clementi

Corrigendum: Adaptive resolution simulation of liquid water

J. Phys.: Condens. Matter 21, 499801, 2009.

**Abstract**

In our two papers [1, 2] , we used a rigid all-atom TIP3P water model with a H-O-H angle of 112.20° instead of, as erroneously stated, the standard rigid TIP3P model [3] with a 104.52° H-O-H angle. All the other parameters are the same as in the standard rigid TIP3P water model. The statistical properties of similarly modified water models have been studied by other authors, cf references [4–6] . Since in the adaptive resolution simulations reported in references [1, 2] the modified TIP3P all-atom water model was studied in combination with the appropriate coarse-grained model, all conclusions of the paper concerning the adaptive resolution simulation remain unaltered.

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**2008**

## Rafael Delgado-Buscalioni, Kurt Kremer, Matej Praprotnik

Concurrent triple-scale simulation of molecular liquids

J. Chem. Phys. 128, 114110, 2008.

**Abstract**

We present a triple-scale simulation of a molecular liquid, in which the atomistic, coarse-grained, and continuum descriptions of the liquid are concurrently coupled. The presented multiscale approach, which covers the length scales ranging from the micro- to macroscale, is a combination of two dual-scale models: a particle-based adaptive resolution scheme (AdResS), which couples the atomic and mesoscopic scales, and a hybrid continuum-molecular dynamics scheme (HybridMD). The combined AdResS-HybridMD scheme successfully sorts out the problem of large molecule insertion in the hybrid particle-continuum simulations of molecular liquids. The combined model is shown to correctly describe the hydrodynamics within a hybrid particle-continuum framework. The presented approach opens up the possibility to perform efficient grand-canonical molecular dynamics simulations of truly open molecular liquid systems.

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## Matej Praprotnik, Luigi Delle Site, Kurt Kremer

Multiscale simulation of soft matter: From scale bridging to adaptive resolution

Annu. Rev. Phys. Chem. 59, 545-571, 2008.

**Abstract**

The relation between atomistic chemical structure, molecular architecture, molecular weight, and material properties is of basic concern in modern soft material science and includes standard properties of bulk materials and surface and interface aspects, as well as the relation between structure and function in nanoscopic objects and molecular assemblies of both synthetic and biological origin. This all implies a thorough understanding on many length and correspondingly time scales, ranging from (sub)atomistic to macroscopic. Presently, computer simulations play an increasingly important, if not central, role. Some problems do not require specific atomistic details, whereas others require them only locally. However, in many cases this strict separation is not sufficient for a comprehensive understanding of systems, and flexible simulation schemes are required that link the different levels of resolution. We here give a general view of the problem regarding soft matter and discuss some specific examples of linked simulation techniques at different resolution levels. We then discuss a recently developed flexible simulation scheme, the AdResS method, which allows one to adaptively change the resolution in certain regions of space on demand.

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## Matej Praprotnik, Christoph Junghans, Luigi Delle Site, Kurt Kremer

Simulation approaches to soft matter: Generic statistical properties vs. chemical details

Comput. Phys. Commun. 179, 51-60, 2008.

**Abstract**

The relation between atomistic structure, architecture, molecular weight and material properties is a basic concern of modern soft material science. This by now goes far beyond standard properties of bulk materials. A typical additional focus is on surface or interface aspects or on the relation between structure and function in nanoscopic molecular assemblies. This all implies a thorough understanding on many length and correspondingly time scales ranging from (sub-)atomic to macroscopic. At this point computer simulations are playing an increasingly important, if not the central role. Traditionally simulations have been separated in two main groups, namely simplified models to deal with generic or universal aspects of polymers, i.e. critical exponents, and those employing classical force field simulations with (almost) all atomistic detail, i.e. for the diffusion of small additives in a small “sample”. Still characteristic problems, which require huge systems and/or long times in combination with a chemistry specific model, cannot be tackled by these methods alone. More recently with the development of scale bridging or multiscale simulation techniques, these different approaches have been combined into an emerging rather powerful tool. It is the purpose of this contribution to give a few examples of how such an approach can be used to understand specific material properties.

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## Silvina Matysiak, Cecilia Clementi, Matej Praprotnik, Kurt Kremer, Luigi Delle Site

Modeling diffusive dynamics in adaptive resolution simulation of liquid water

J. Chem. Phys. 128, 024503, 2008.

**Abstract**

We present a dual-resolution molecular dynamics (MD) simulation of liquid water employing a recently introduced Adaptive Resolution Scheme (AdResS). The spatially adaptive molecular resolution procedure allows for changing from a coarse-grained to an all-atom representation and vice versa on-the-fly. In order to find the most appropriate coarse-grained watermodel to be employed with AdResS, we first study the accuracy of different coarse-grained water models in reproducing the structural properties of the all-atom system. Typically, coarse-grained molecular models have a higher diffusion constant than the corresponding all-atom models due to the reduction in degrees of freedom (DOFs) upon coarse-graining that eliminates the fluctuating forces associated with those integrated-out molecular DOFs. Here, we introduce the methodology to obtain the same diffusionaldynamics across different resolutions. We show that this approach leads to the correct description of the here relevant structural, thermodynamical, and dynamical properties, i.e., radial distribution functions, pressure, temperature, and diffusion, of liquid water at ambient conditions.

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## Christoph Junghans, Matej Praprotnik, Kurt Kremer

Transport properties controlled by a thermostat: An extended dissipative particle dynamics thermostat

Soft Matter 4, 156-161, 2008.

**Abstract**

We introduce a variation of the dissipative particle dynamics (DPD) thermostat that allows for controlling transport properties of molecular fluids. The standard DPD thermostat acts only on a relative velocity along the interatomic axis. Our extension includes the damping of the perpendicular components of the relative velocity, whilst keeping the advantages of conserving Galilei invariance and within our error bar also hydrodynamics. This leads to a second friction parameter for tuning the transport properties of the system. Numerical simulations of a simple Lennard-Jones fluid and liquid water demonstrate a very sensitive behaviour of the transport properties, e.g., viscosity, on the strength of the new friction parameter. We envisage that the new thermostat will be very useful for the coarse-grained and adaptive resolution simulations of soft matter, where the diffusion constants and viscosities of the coarse-grained models are typically too high/low, respectively, compared to all-atom simulations.

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## Matej Praprotnik, Stanko Hočevar, Milan Hodošček, Matej Penca, Dušanka Janežič

New all-atom force field for molecular dynamics simulation of AlPO4-34 molecular sieve

J. Comput. Chem. 29, 122-129, 2008.

**Abstract**

A force field of the triclinic framework of AlPO4-34, important in methanol–hydrocarbon conversion reactions, was developed using an empirical potential function. Molecular dynamics simulation of an AlPO4-34 triclinic framework segment of 1216 atoms, containing the template molecules isopropylamine and water, was performed with explicit consideration of atomic charges. The average RMS difference between instantaneous positions of the framework atoms during 1 ns simulation and their positions in the structure determined from single crystal X-ray diffraction was calculated, and the average structure of the flexible framework was determined. The computed Debye-Waller factors and simulated FTIR spectra are in good agreement with the experimental data. The new force field permits detailed molecular dynamics simulations of flexible, charged aluminophosphate molecular sieves which should lead to a better understanding of the catalytic processes and the crucial role played by templating molecules.

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**2007**

## Matej Praprotnik, Silvina Matysiak, Luigi Delle Site, Kurt Kremer, Cecilia Clementi

Adaptive resolution simulation of liquid water

J. Phys.: Condens. Matter 19, 292201, 2007.

**Abstract**

Water plays a central role in biological systems and processes, and is equally relevant in a large range of industrial and technological applications. Being the most important natural solvent, its presence uniquely influences biological function as well as technical processes. Because of their importance, aqueous solutions are among the most experimentally and theoretically studied systems. However, many questions still remain open. Both experiments and theoretical models are usually restricted to specific cases. In particular all-atom simulations of biomolecules and materials in water are computationally very expensive and often not possible, mainly due to the computational effort to obtain water–water interactions in regions not relevant for the problem under consideration. In this paper we present a coarse-grained model that can reproduce the behaviour of liquid water at a standard temperature and pressure remarkably well. The model is then used in a multiscale simulation of liquid water, where a spatially adaptive molecular resolution procedure allows one to change from a coarse-grained to an all-atom representation on-the-fly. We show that this approach leads to the correct description of essential thermodynamic and structural properties of liquid water. Our adaptive multiscale scheme allows for significantly greater extensive simulations than existing approaches by taking explicit water into account only in the regions where the atomistic details are physically relevant.

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## Matej Praprotnik, Luigi Delle Site, Kurt Kremer

A macromolecule in a solvent: Adaptive resolution molecular dynamics simulation

J. Chem. Phys. 126, 134902, 2007.

**Abstract**

The authors report adaptive resolution molecular dynamics simulations of a flexible linear polymer in solution. The solvent, i.e., a liquid of tetrahedral molecules, is represented within a certain radius from the polymer’s center of mass with a high level of detail, while a lower coarse-grained resolution is used for the more distant solvent. The high resolution sphere moves with the polymer and freely exchanges molecules with the low resolution region through a transition regime. The solvent molecules change their resolution and number of degrees of freedom on the fly. The authors show that their approach correctly reproduces the static and dynamic properties of the polymer chain and surrounding solvent.

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## Matej Praprotnik, Kurt Kremer, Luigi Delle Site

Fractional dimensions of phase space variables: A tool for varying the degrees of freedom of a system in a multiscale treatment

J. Phys. A: Math. Theor. 40, F281-F288, 2007.

**Abstract**

We show how the idea of fractal dimensions of phase space variables can be employed to develop a concept of adaptive resolution treatment of a molecular liquid. The resulting theoretical framework allows for calculation of statistical averages of thermodynamic quantities in multiresolution computer simulation algorithms where the molecular degrees of freedom change on the fly.

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## Matej Praprotnik, Kurt Kremer, Luigi Delle Site

Adaptive molecular resolution via a continous change of the phase space dimensionality

Phys. Rev. E 75, 017701, 2007.

**Abstract**

For the study of complex synthetic and biological molecular systems by computer simulations one is still restricted to simple model systems or by far too small time scales. To overcome this problem multiscale techniques are being developed. However, in almost all cases, the regions and molecules of different resolution are kept fixed and are not in equilibrium with each other. We here give a basic theoretical framework for an efficient and flexible coupling of the different regimes. The approach leads to a concept, which can be seen as a geometry-induced phase transition, and to a counterpart of the equipartition theorem for fractional degrees of freedom. This represents the initial step in developing a general theoretical framework for computer simulation methods applying simultaneously different levels of resolution.

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**2006**

## Matej Praprotnik, Luigi Delle Site, Kurt Kremer

Adaptive resolution scheme for efficient hybrid atomistic-mesoscale molecular dynamics simulations of dense liquids

Phys. Rev. E 73, 066701, 2006.

**Abstract**

The adaptive resolution scheme (AdResS) for efficient hybrid particle-based atomistic/mesoscale molecular dynamics (MD) simulations recently introduced by us, [J. Chem. Phys. 123, 224106 (2005)] is extended to high density molecular liquids with spherical boundaries between the atomistic and mesoscale regions. The key feature of this approach is that it allows for a dynamical change of the number of molecular degrees of freedom during the course of a MD simulation by an on-the-fly switching between the atomistic and mesoscopic levels of detail. Pressure and density variations occurring at the atomistic/mesoscale boundary in the original version are considerably reduced employing the improved methodology presented here.

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**2005**

## Matej Praprotnik, Luigi Delle Site, Kurt Kremer

Adaptive resolution molecular-dynamics simulation: Changing the degrees of freedom on the fly

J. Chem. Phys. 123, 224106, 2005.

**Abstract**

We present a new adaptive resolution technique for efficient particle-based multiscale molecular-dynamics simulations. The presented approach is tailor-made for molecular systems where atomistic resolution is required only in spatially localized domains whereas a lower mesoscopic level of detail is sufficient for the rest of the system. Our method allows an on-the-fly interchange between a given molecule’s atomic and coarse-grained levels of description, enabling us to reach large length and time scales while spatially retaining atomistic details of the system. The new approach is tested on a modelsystem of a liquid of tetrahedral molecules. The simulation box is divided into two regions: one containing only atomistically resolved tetrahedral molecules, and the other containing only one-particle coarse-grained spherical molecules. The molecules can freely move between the two regions while changing their level of resolution accordingly. The hybrid and the atomistically resolved systems have the same statistical properties at the same physical conditions.

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## Matej Praprotnik, Dušanka Janežič

Molecular dynamics integration meets standard theory of molecular vibrations

J. Chem. Inf. Model. 45, 1571-1579, 2005.

**Abstract**

An iterative SISM (split integration symplectic method) for molecular dynamics (MD) integration is described. This work explores an alternative for the internal coordinate system prediction in the SISM introduced by Janežič et al. (J. Chem. Phys. 2005, 122, 174101). The SISM, which employes a standard theory of molecular vibrations, analytically resolves the internal high-frequency molecular vibrations. This is accomplished by introducing a translating and rotating internal coordinate system of a molecule and calculating normal modes of an isolated molecule only. The Eckart frame, which is usually used in the standard theory of molecular vibrations as an internal coordinate system of a molecule, is adopted to be used within the framework of the second order generalized leapfrog scheme. In the presented MD integrator the internal coordinate frame at the end of the integration step is predicted halfway through the integration step using a predictor-corrector type iterative approach thus ensuring the method’s time reversibility. The iterative SISM, which is applicable to any system of molecules with one equilibrium configuration, was applied here to perform all-atom MD simulations of liquid CO2 and SO2. The simulation results indicate that for the same level of accuracy, this algorithm allows significantly longer integration time steps than the standard second-order leapfrog Verlet (LFV) method.

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## Dušanka Janežič, Matej Praprotnik, Franci Merzel

Molecular dynamics integration and molecular vibrational theory. I. New symplectic integrators

J. Chem. Phys. 122, 174101, 2005.

**Abstract**

New symplectic integrators have been developed by combining molecular dynamics integration with the standard theory of molecular vibrations to solve the Hamiltonian equations of motion. The presented integrators analytically resolve the internal high-frequency molecular vibrations by introducing a translating and rotating internal coordinate system of a molecule and calculating normal modes of an isolated molecule only. The translation and rotation of a molecule are treated as vibrational motions with the vibrational frequency zero. All types of motion are thus described in terms of the normal coordinates. The method’s time reversibility requirement was used to determine the equations of motion for internal coordinate system of a molecule. The calculation of long-range forces is performed numerically within the generalized second-order leap-frog scheme, in the same way as in standard second-order symplectic methods. The new methods for integrating classical equations of motion using normal modeanalysis allow us to use a long integration step and are applicable to any system of molecules with one equilibrium configuration.

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## Matej Praprotnik, Dušanka Janežič

Molecular dynamics integration and molecular vibrational theory. II. Simulation of nonlinear molecules

J. Chem. Phys. 122, 174102, 2005.

**Abstract**

A series of molecular dynamics (MD) simulations of nonlinear molecules has been performed to test the efficiency of newly introduced semianalytical second-order symplectic time-reversible MD integrators that combine MD and the standard theory of molecular vibrations. The simulation results indicate that for the same level of accuracy, the new algorithms allow significantly longer integration time steps than the standard second-order symplectic leap-frog Verlet method. Since the computation cost per integration step using new MD integrators with longer time steps is approximately the same as for the standard method, a significant speed-up in MD simulation is achieved.

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## Matej Praprotnik, Dušanka Janežič

Molecular dynamics integration and molecular vibrational theory. III. The infrared spectrum of water

J. Chem. Phys. 122, 174103, 2005.

**Abstract**

The new symplectic molecular dynamics (MD) integrators presented in the first paper of this series were applied to perform MD simulations of water. The physical properties of a system of flexible TIP3P water molecules computed by the new integrators, such as diffusion coefficients, orientation correlation times, and infrared (IR) spectra, are in good agreement with results obtained by the standard method. The comparison between the new integrators’ and the standard method’s integration time step sizes indicates that the resulting algorithm allows a 3.0fs long integration time step as opposed to the standard leap-frog Verlet method, a sixfold simulation speed-up. The accuracy of the method was confirmed, in particular, by computing the IR spectrum of water in which no blueshifting of the stretching normal mode frequencies is observed as occurs with the standard method.

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**2004**

## Matej Praprotnik, Dušanka Janežič, Janez Mavri

Temperature dependence of water vibrational spectrum: A molecular dynamics simulation study

J. Phys. Chem. A 108, 11056-11062, 2004.

**Abstract**

Vibrational spectroscopy studies show that the bulk water bending band becomes narrower with increasing temperature (Maréchal, Y. J. Mol. Struct. 1994, 322, 105). Since this counterintuitive effect is not associated with the quantum nature of nuclear motion a molecular dynamics (MD) simulation is expected to reproduce it even in the classical limit. We have performed a classical MD simulation of the flexible simple point charge (SPC) and extended SPC (SPC/E) water models to determine the temperature dependence of the bulk water vibrational spectrum. The intramolecular water potential proposed by Toukan and Rahman, including a stretch−bend coupling term, was applied. We performed MD simulations at −4 and 80 °C to compare the calculated vibrational spectra, in particular, the band associated with the bending mode, with the experiment. The experimentally determined narrowing of the bending band with increasing temperature is not reproducible by MD simulation with the applied force field. However, the results show that this approach successfully reproduces all other experimentally observed spectroscopic properties of bulk water.

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## Roman Trobec, Marjan Šterk, Matej Praprotnik, Dušanka Janežič

Parallel programming library for molecular dynamics simulations

Int. J. Quant. Chem. 96, 530-536, 2004.

**Abstract**

A parallel programming library for molecular dynamics (MD) simulations is described and applied to the recently proposed split integration symplectic method (SISM) for MD simulation. The results show that for a system of 1024 linear chain molecules with an integration step of 4.5 fs parallel execution of SISM with the particle–particle interactions (PPIs) library on 32 computers gives efficiency of 95.6%. The results also show the parallel simulation of n particles is scalable with the number of processors p and the time requirement is proportional to n2/p for n/p large enough, which guarantees optimal speed-up.

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## Matej Praprotnik, Marjan Šterk, Roman Trobec

Inhomogeneous heat-conduction problems solved by a new explicit finite difference scheme

IJPAM 13, 275-291, 2004.

**Abstract**

A heat conduction in systems composed of biomaterials, such as the heart muscle, is described by the familiar heat conduction equation. Due to the inhomogeneity of these materials the equations defining the diffusion problem are difficult to solve. A new explicit finite difference scheme for solving the heat conduction equation for inhomogeneous materials is derived. The new scheme has the same computational complexity as the standard scheme and gives the same solution but with increased resolution of the temperature grid. It was derived and studied on a simple one dimensional problem of heat conduction and applied to studying the temperature distribution in a three dimensional model of the heart muscle.

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**2003**

## Dušanka Janežič, Matej Praprotnik

Molecular dynamics integration time step dependence of the split integration symplectic method on system density

J. Chem. Inf. Comput. Sci. 43, 1922-1927, 2003.

**Abstract**

This paper shows that the maximal size of the integration time step of the Split Integration Symplectic Method (SISM) for molecular dynamics (MD) integration, a combination of the analytical solution of the high-frequency harmonic part of the Hamiltonian and the numerical solution of the low-frequency remaining part, depends on the system density. This approach was tested on a system of linear chain molecules. The numerical results indicate that the integration time step used by the SISM is limited by atoms’ motion generated by the electrostatic and Lennard-Jones interactions in the system. As the density of the system increases, the size of the integration time step allowed by the SISM thus becomes smaller but remains significantly larger than possible by standard methods of the same order and complexity.

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**2002**

## Marjan Šterk, Roman Trobec, Matej Praprotnik

Numerical schemes for fluid flow and heat transfer in medical simulations

Parallel and Distributed Computing Practices 5, 321-329, 2002.

**Abstract**

Incompressible fluid flow is governed by the Navier-Stokes equation, which, together with the diffusion and continuity equations, forms a coupled system of partial differential equations that have to be solved to simulate the fluid dynamics. We describe a finite difference scheme and boundary conditions used to solve the system of partial differential equations on general 3-dimensional domains with explicit integration in time. The most computationally intensive part is the pressure equation that requires the solution of a sparse linear system in each time-step of the simulation. Various iterative methods for the solution of the linear system are tested and compared among which the multigrid method outperforms others. Some test examples are given to prove the validity of the simulation results. The paper concludes with an analysis of parallel computational complexity of the SOR method and parallelization strategy for the multigrid method.

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## Matej Praprotnik, Dušanka Janežič

The Split Integration Symplectic Method

Cell. Mol. Biol. Lett. 7, 147-148, 2002.

**Abstract**

The split integration symplectic method (SISM) for Hamiltonian systems based on factorization of the Liouvillepropagator is presented.

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**2001**

## Roman Trobec, Marjan Šterk, Matej Praprotnik, Dušanka Janežič

Implementation and evaluation of MPI-based parallel MD program

Int. J. Quant. Chem. 84, 23-31, 2001.

**Abstract**

The message-passing interface (MPI)-based object-oriented particle–particle interactions (PPI) library is implemented and evaluated. The library can be used in the n-particle simulation algorithm designed for a ring of p interconnected processors. The parallel simulation is scalable with the number of processors, and has the time requirement proportional to n2/p if n/p is large enough, which guarantees optimal speedup. In a certain range of problem sizes, the speedup becomes superlinear because enough cache memory is available in the system. The library is used in a simple way by any potential user, even with no deep programming knowledge. Different simulations using particles can be implemented on a wide spectrum of different computer platforms. The main purpose of this article is to test the PPI library on well-known methods, e.g., the parallel molecular dynamics (MD) simulation of the monoatomic system by the second-order leapfrog Verlet algorithm. The performances of the parallel simulation program implemented with the proposed library are competitive with a custom-designed simulation code. Also, the implementation of the split integration symplectic method, based on the analytical calculation of the harmonic part of the particle interactions, is shown, and its expected performances are predicted.

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## Dušanka Janežič, Matej Praprotnik

Symplectic molecular dynamics integration using normal mode analysis

Int. J. Quant. Chem. 84, 2-12, 2001.

**Abstract**

The split integration symplectic method (SISM) for molecular dynamics (MD) integration using normal mode analysis based on a factorization of the Liouville propagator is presented. This approach is quite distinct from others that use fractional-step methods, owing to the analytical treatment of high-frequency motions. The method involves splitting the total Hamiltonian of the system into a harmonic part and the remaining part. Then the Hamilton equations are solved using a second-order generalized leapfrog integration scheme in which the purely harmonic Hamiltonian (which represents the main contribution of the chemical bonds and angles) is treated analytically, i.e., independent of the step size of integration, by a normal mode analysis that is carried out only once, at the beginning of calculation. The whole integration step combines analytical evolution of the harmonic part of the Hamiltonian with a correction arising from the remaining part. The proposed algorithm requires only one force evaluation per integration step. The algorithm was tested on a simple system of linear chain molecules. Results demonstrate the method makes possible the integration of the MD equations over larger time steps without loss of stability while being economical in computer time.

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## Books & Book Chapters

## Matej Praprotnik, Robin Cortes-Huerto, Raffaello Potestio, Luigi Delle Site

Adaptive resolution molecular dynamics technique

In: W. Andreoni, S. Yip (Eds.) Handbook of materials modeling. Volume 1 Methods: Theory and modeling, pp. 1443-1457, Springer, Cham, 2020.

**Abstract**

Soft matter systems display properties that span different time and length scales. In addition, scales’ interplay is often the key to understand fundamental mechanisms to the aim of controlling and/or designing materials with properties on demand. On the other hand, computational soft matter is limited by computational power for both, size and time of simulation and analysis of large sets of data. In this perspective, computational efficiency to treat large systems on long time scales becomes one of the main goals in constructing modern algorithms, together with the capability of designing theoretical schemes for data analysis capable of extracting the relevant information of interest above all the effects of scales’ interplay. One common and recurrent feature, in such studies, is the need to include relevant chemical details in a specific region where an event of interest is taking place, while the environment plays simply the role of a macroscopic thermodynamic bath that can be treatable at a coarse-grained level. Thus, an efficient computational strategy consists in employing multiple resolution methods, which simultaneously consider models with different resolution in different regions. This chapter provides a basic introduction to the adaptive resolution simulation (AdResS) method and its recent extensions. This methodology is designed with the idea of efficient computation and analysis of multiple scales as envisaged above. We will report its basic principles and technical aspects for the various directions along which the original idea was developed. As it will emerge in the next sections, the basic idea of adaptive resolution, already highly efficient in its first implementation, has now reached a high level of theoretical solidity, being framed in different but complementary ways in physically rigorous principles. Finally, selected applications, relevant in the field of materials science, chemical physics, and biochemistry, are illustrated in order to show the advanced possibilities of application of the method.

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## Xin Bian, Matej Praprotnik

Domain decomposition methods for multiscale modeling

In: W. Andreoni, S. Yip (Eds.) Handbook of materials modeling. Volume 2 Applications: Current and emerging materials, pp. 2551-2571, Springer, Cham, 2020.

**Abstract**

Domain decomposition methods (DDM), which originate from the Schwarz alternating method to solve elliptic partial different equations, are largely extended and prove to have increasing influences on multiscale modeling of materials. We discuss some of the important extensions of the DDM in the fields of multiscale modeling for soft materials such as simple and complex fluids. To this end, we typically model the fluids in two or more levels of detail, which exploits the computational efficiency of the coarse model and physical accuracy of the fine description. For simple fluids, we take a continuum perspective to couple the molecular dynamics (MD) and Navier-Stokes equations by matching the state variables and/or fluxes across the hybrid interface. For complex fluids, we take a discrete perspective to encompass the complex structure of the molecules and couple the MD with coarse-grained MD by interpolating the forces between the two levels of descriptions.

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## Members of the PRACE Scietific Steering Committee

Scientific case

Erik Lindahl (Editor-in-chief) The scientific case for computing in Europe 2018 – 2026, Insight Publishers, Bristol, 2018.

The third version of the scientific case for computing in Europe by the PRACE Scientific Steering Committee.

## Matej Praprotnik, Luigi Delle Site

Multiscale molecular modeling

In: Luca Monticelli, Emppu Salonen (Eds.) Biomolecular simulations. Methods in molecular biology (Methods and protocols), vol. 924, pp. 567-583, Humana Press, Totowa, NJ, 2013.

**Abstract**

We review the basic theoretical principles of the adaptive resolution simulation scheme (AdResS). This method allows to change molecular resolution on-the-fly during a simulation by changing the number of degrees of freedom in specific regions of space where the required resolution is higher than in the rest of the system. We also report about recent extensions of the method to the continuum and quantum regimes.

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## Dušanka Janežič, Urban Borštnik, Matej Praprotnik

Parallel approaches in molecular dynamics simulations

In: R. Trobec, M. Vajteršic, P. Zinterhof (Eds.) Parallel computing: Numerics, applications, and trends, pp. 281-305, Springer, London, 2009.

**Abstract**

In this contribution we will present the survey of our past and current endeavor on parallel approaches in molecular modeling algorithm development, for example, molecular dynamics (MD) simulation. In particular, we will describe the new split integration symplectic method for the numerical solution of molecular dynamics equations and methods for the determination of vibrational frequencies and normal modes of large systems, and the distributed diagonal force decomposition method, a parallel method for MD simulation.

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## Christoph Junghans, Matej Praprotnik, Luigi Delle Site

Adaptive resolution schemes

In: J. Grotendorst, N. Attig, S. Blügel, D. Marx (Eds.) Multiscale simulation methods in molecular sciences, NIC series, vol. 42, pp. 359-379, Institute for Advanced Simulation, Forschungszentrum Jülich, 2009.

**Abstract**

The Adaptive Resolution Scheme (AdResS) is a simulation method, which allows to perform Molecular Dynamics (MD) simulations treating different regions with different molecular resolutions. The different scales are coupled within a unified approach by changing the number of degrees of freedom on the fly and preserving the free exchange of particles between regions of different resolution. Here we describe the basic physical principles of the algorithm and illustrate some of its relevant applications.

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