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    2016

  • Serena Donnini, R. Thomas Ullmann, Gerrit Groenhof, Helmut Grubmüller; 2016; Charge-neutral constant pH molecular dynamics simulations using a parsimonious proton buffer; vol. 12; p. 1040-1051; doi 10.1021/acs.jctc.5b01160
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    Abstract: In constant pH molecular dynamics simulations, the protonation states of titratable sites can respond to changes of the pH and of their electrostatic environment. Consequently, the number of protons bound to the biomolecule, and therefore the overall charge of the system, fluctuates during the simulation. To avoid artifacts associated with a non-neutral simulation system, we introduce an approach to maintain neutrality of the simulation box in constant pH molecular dynamics simulations, while maintaining an accurate description of all protonation fluctuations. Specifically, we introduce a proton buffer that, like a buffer in experiment, can exchange protons with the biomolecule enabling its charge to fluctuate. To keep the total charge of the system constant, the uptake and release of protons by the buffer are coupled to the titration of the biomolecule with a constraint. We find that, because the fluctuation of the total charge (number of protons) of a typical biomolecule is much smaller than the number of titratable sites of the biomolecule, the number of buffer sites required to maintain overall charge neutrality without compromising the charge fluctuations of the biomolecule, is typically much smaller than the number of titratable sites, implying markedly enhanced simulation and sampling efficiency.
    BibTeX entry:
    @Article{Donnini2016,
     author = "Serena Donnini and R. Thomas Ullmann  and Gerrit Groenhof and Helmut Grubm{\"{u}}ller",
     title = "{Charge-neutral constant pH molecular dynamics simulations using a parsimonious proton buffer}",
     journal = "J. Chem. Theory Comput.",
     volume = "12",
     pages = "1040-1051",
     year = "2016",
     doi = "10.1021/acs.jctc.5b01160",
     abstract = "{In constant pH molecular dynamics simulations, the protonation states of titratable
                  sites can respond to changes of the pH and of their electrostatic environment.
                  Consequently, the number of protons bound to the biomolecule, and therefore the
                  overall charge of the system, fluctuates during the simulation. To avoid artifacts
                  associated with a non-neutral simulation system, we introduce an approach to maintain
                  neutrality of the simulation box in constant pH molecular dynamics simulations, while
                  maintaining an accurate description of all protonation fluctuations. Specifically,
                  we introduce a proton buffer that, like a buffer in experiment, can exchange protons
                  with the biomolecule enabling its charge to fluctuate. To keep the total charge of the
                  system constant, the uptake and release of protons by the buffer are coupled to the
                  titration of the biomolecule with a constraint. We find that, because the fluctuation
                  of the total charge (number of protons) of a typical biomolecule is much smaller than
                  the number of titratable sites of the biomolecule, the number of buffer sites required
                  to maintain overall charge neutrality without compromising the charge fluctuations of
                  the biomolecule, is typically much smaller than the number of titratable sites, implying
                  markedly enhanced simulation and sampling efficiency.}",
    }
    
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    2012

  • R. Thomas Ullmann, G. Matthias Ullmann; 2012; GMCT: A Monte Carlo simulation package for macromolecular receptors; vol. 33; p. 887-900; doi 10.1002/jcc.22919
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    Abstract: GMCT (generalized Monte Carlo titration) is a versatile suite of computer programs for the efficient simulation of complex macromolecular receptor systems as for example proteins. The computational model of the system is based on a microstate description of the receptor and an average description of its surroundings in terms of chemical potentials. The receptor can be modeled in great detail including conformational flexibility and many binding sites with multiple different forms that can bind multiple different ligand types. Membrane embedded systems can be modeled with electrochemical potential gradients. Overall properties of the receptor as well as properties of individual sites can be studied with a variety of different Monte Carlo (MC) simulation methods. Metropolis MC, Wang-Landau MC and efficient free energy calculation methods are included. GMCT is distributed as free open source software at www.bisb.uni-bayreuth.de under the terms of the GNU Affero General Public License.
    BibTeX entry:
    @Article{Ullmann2012GMCT,
      author   = "R. Thomas Ullmann and G. Matthias Ullmann",
      title    = "{GMCT: A Monte Carlo simulation package for
                   macromolecular receptors}",
      journal  = "J. Comput. Chem.",
      volume   = "33",
      pages    = "887-900",
      year     = 2012,
      doi      = {10.1002/jcc.22919},
      abstract = "{GMCT (generalized Monte Carlo titration) is a versatile
                   suite of computer programs for the efficient
                   simulation of complex macromolecular receptor systems as
                   for example proteins.
                   The computational model of the system is based on a
                   microstate description of the receptor and an average
                   description of its surroundings in terms of chemical
                   potentials.
                   The receptor can be modeled in great detail including
                   conformational flexibility and many binding sites with
                   multiple different forms that can bind multiple different
                   ligand types.
                   Membrane embedded systems can be modeled with
                   electrochemical potential gradients.
                   Overall properties of the receptor as well as properties
                   of individual sites can be studied with a variety of
                   different Monte Carlo (MC) simulation methods.
                   Metropolis MC, Wang-Landau MC and efficient free energy
                   calculation methods are included.
                   GMCT is distributed as free open source software at
                   www.bisb.uni-bayreuth.de under the terms of the
                   GNU Affero General Public License.
                   }",
    }
    
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  • R. Thomas Ullmann; 2012; Monte Carlo simulation methods for studying the thermodynamics of ligand binding & transfer processes in biomolecules; PhD thesis; Universitaet Bayreuth
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    Abstract: p> The binding and transfer of ligands is of central importance for the function of many biomolecular systems. The main topic of this thesis is the development and application of Monte Carlo (MC) simulation methods for studying complex ligand binding equilibria which can also involve conformational changes. The simulated systems were described by microstates within a continuum electrostatics / molecular mechanics (CE/MM) model of the receptor-ligand system. The CE/MM modeling methodology was improved. The improvements led to more detailed molecular models that enable a more realistic reproduction of system properties and environmental conditions. The developed simulation methods were applied to biomolecular systems whose function involves aspects that are important for the understanding of bioenergetic energy transduction. The results of this thesis are presented in five articles that are published in peer reviewed scientific journals.

    Manuscript A presents the Monte Carlo simulation software GMCT which was largely developed in this thesis. The software offers a variety of different simulation methods that allow the user to harness the full potential of CE/MM models in the simulation of complex receptor systems.

    Manuscript B presents a novel theoretical framework for free energy calculations with the free energy perturbation method. The novel framework is more broadly applicable and can lead to more efficient simulations than previous formulations. The derivation of the formalism also led to interesting insights into general statistical mechanics. The formalism was implemented in GMCT and could already be used fruitfully for the free energy calculations presented in Manuscripts C and D.

    Manuscript C demonstrates the application of free energy measures of cooperativity to study the coupling of protonation, reduction and conformational change in azurin from Pseudomonas aeruginosa (PaAz). Such a coupling is prototypic for bioenergetic systems because it forms the thermodynamic basis of their energy transducing function. PaAz is an experimentally well characterized, small electron transport protein. For this reason, PaAz was used here as model system to demonstrate the usefulness of cooperativity free energies in detecting and quantifying thermodynamic coupling between events in complex biomolecular systems. The results of this study led to new insight that could help to determine the still enigmatic physiological role of PaAz.

    In Manuscript D, free energy calculations were applied to study the thermodynamics of transport through the ammonium transporter Amt-1 from Archaeoglobus fulgidus (AfAmt-1). Ammonium is the most directly utilizable nitrogen source for plants and microorganisms. AfAmt-1 and its homologues facilitate the transport of ammonia / ammonium across biological membranes in living beings from all domains of life. It is intensely debated how these proteins perform their function and whether ammonia or its protonated form ammonium is actually transported. The study extended upon previous theoretical studies by including the effects of substrate concentration, electrochemical transmembrane gradients, proton-coupled binding equilibria and competitive binding of different ligand species. It was found that the transported species is most likely the ammonium ion. An NH3}/H+ symport mechanism that involves a pair of coplanar histidine residues at the center of the transmembrane pore as transient proton acceptor is made plausible by the high genetic conservation of these residues.

    Manuscript E presents a first application of the microstate description within a CE/MM model to the simulation of the non-equilibrium dynamics of a molecular system. We simulated the re-reduction kinetics of primary electron donor in the photocycle of the bacterial photosynthetic reaction center from Blastochloris viridis. The simulation results are in very good agreement with experimentally measured data.
    BibTeX entry:
    @PhDthesis{Ullmann2012PhD,
      author   = "R. Thomas Ullmann",
      title    = "Monte Carlo simulation methods for studying the
                  thermodynamics of ligand binding & transfer processes
                  in biomolecules",
      school   = "Universit{\"{a}}t Bayreuth",
      year     = 2012,
      abstract = "{
                   

    The binding and transfer of ligands is of central importance for the function of many biomolecular systems. The main topic of this thesis is the development and application of Monte Carlo (MC) simulation methods for studying complex ligand binding equilibria which can also involve conformational changes. The simulated systems were described by microstates within a continuum electrostatics / molecular mechanics (CE/MM) model of the receptor-ligand system. The CE/MM modeling methodology was improved. The improvements led to more detailed molecular models that enable a more realistic reproduction of system properties and environmental conditions. The developed simulation methods were applied to biomolecular systems whose function involves aspects that are important for the understanding of bioenergetic energy transduction. The results of this thesis are presented in five articles that are published in peer reviewed scientific journals.

    Manuscript A presents the Monte Carlo simulation software GMCT which was largely developed in this thesis. The software offers a variety of different simulation methods that allow the user to harness the full potential of CE/MM models in the simulation of complex receptor systems.

    Manuscript B presents a novel theoretical framework for free energy calculations with the free energy perturbation method. The novel framework is more broadly applicable and can lead to more efficient simulations than previous formulations. The derivation of the formalism also led to interesting insights into general statistical mechanics. The formalism was implemented in GMCT and could already be used fruitfully for the free energy calculations presented in Manuscripts C and D.

    Manuscript C demonstrates the application of free energy measures of cooperativity to study the coupling of protonation, reduction and conformational change in azurin from Pseudomonas aeruginosa (PaAz). Such a coupling is prototypic for bioenergetic systems because it forms the thermodynamic basis of their energy transducing function. PaAz is an experimentally well characterized, small electron transport protein. For this reason, PaAz was used here as model system to demonstrate the usefulness of cooperativity free energies in detecting and quantifying thermodynamic coupling between events in complex biomolecular systems. The results of this study led to new insight that could help to determine the still enigmatic physiological role of PaAz.

    In Manuscript D, free energy calculations were applied to study the thermodynamics of transport through the ammonium transporter Amt-1 from Archaeoglobus fulgidus (AfAmt-1). Ammonium is the most directly utilizable nitrogen source for plants and microorganisms. AfAmt-1 and its homologues facilitate the transport of ammonia / ammonium across biological membranes in living beings from all domains of life. It is intensely debated how these proteins perform their function and whether ammonia or its protonated form ammonium is actually transported. The study extended upon previous theoretical studies by including the effects of substrate concentration, electrochemical transmembrane gradients, proton-coupled binding equilibria and competitive binding of different ligand species. It was found that the transported species is most likely the ammonium ion. An NH3}$/H$^{+} symport mechanism that involves a pair of coplanar histidine residues at the center of the transmembrane pore as transient proton acceptor is made plausible by the high genetic conservation of these residues.

    Manuscript E presents a first application of the microstate description within a CE/MM model to the simulation of the non-equilibrium dynamics of a molecular system. We simulated the re-reduction kinetics of primary electron donor in the photocycle of the bacterial photosynthetic reaction center from Blastochloris viridis. The simulation results are in very good agreement with experimentally measured data.}", }
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  • R. Thomas Ullmann, Susana L. A. Andrade, G. Matthias Ullmann; 2012; Thermodynamics of transport through the ammonium transporter Amt-1 investigated with free energy calculations; J. Phys. Chem. B; vol. 116; p. 9690-9703; doi 10.1021/jp305440f
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    Abstract: Amt-1 from Archaeoglobus fulgidus (AfAmt-1) belongs to the Amt/Rh family of ammonium/ammonia transporting membrane proteins. The transport mode and the precise microscopic permeation mechanism utilized by these proteins are intensely debated. Open questions concern the identity of the transported substrate (ammonia and/or ammonium) and whether the transport is passive or active. To address these questions, we studied the overall thermodynamics of the different transport modes as a function of the environmental conditions. Then, we investigated the thermodynamics of the underlying microscopic transport mechanisms with free energy calculations within a continuum electrostatics model. The formalism developed for this purpose is of general utility in the calculation of binding free energies for ligands with multiple protonation forms or other binding forms. The results of our calculations are compared to the available experimental and theoretical data on Amt/Rh proteins and discussed in light of the current knowledge on the physiological conditions experienced by microorganisms and plants. We found that microscopic models of electroneutral and electrogenic transport modes are in principle thermodynamically viable. However, only the electrogenic variants have a net thermodynamic driving force under the physiological conditions experienced by microorganisms and plants. Thus, the transport mechanism of AfAmt-1 is most likely electrogenic.
    BibTeX entry:
    @Article{Ullmann2012Amt-1,
      author   = "R. Thomas Ullmann and Susana L. A. Andrade and G. Matthias Ullmann",
      title    = "{Thermodynamics of transport through the ammonium transporter Amt-1
                   investigated with free energy calculations}",
      journal  = jpcB,
      volume   = {116},
      pages    = {9690-9703},
      year     = 2012,
      doi      = {10.1021/jp305440f},
      abstract = "{Amt-1 from Archaeoglobus fulgidus (AfAmt-1) belongs
                   to the Amt/Rh family of ammonium/ammonia transporting membrane
                   proteins.
                   The transport mode and the precise microscopic permeation
                   mechanism utilized by these proteins are intensely debated.
                   Open questions concern the identity of the transported
                   substrate (ammonia and/or ammonium) and whether the
                   transport is passive or active.
                   To address these questions, we studied the overall
                   thermodynamics of the different transport modes as
                   a function of the environmental conditions.
                   Then, we investigated the thermodynamics of the
                   underlying microscopic transport mechanisms with
                   free energy calculations within a continuum
                   electrostatics model.
                   The formalism developed for this purpose is
                   of general utility in the calculation of
                   binding free energies for ligands with multiple
                   protonation forms or other binding forms.
                   The results of our calculations are compared
                   to the available experimental and theoretical
                   data on Amt/Rh proteins and
                   discussed in light of the current knowledge
                   on the physiological conditions
                   experienced by microorganisms and plants.
                   We found that microscopic models of electroneutral
                   and electrogenic transport modes are in principle
                   thermodynamically viable.
                   However, only the electrogenic variants have a net
                   thermodynamic driving force under the physiological
                   conditions experienced by microorganisms and plants.
                   Thus, the transport mechanism of AfAmt-1 is most
                   likely electrogenic.}",
    }
    
    Files:
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    main paper 1pdf
    supporting information 1pdf

    2011

  • R. Thomas Ullmann, G. Matthias Ullmann; 2011; Coupling of protonation, reduction and conformational change in azurin from Pseudomonas aeruginosa investigated with free energy measures of cooperativity; J. Phys. Chem. B; vol. 115; p. 10346-10359; doi 10.1021/jp204644h
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    Abstract: We used free energy calculations within a continuum electrostatics model to analyze the coupling of protonation, reduction and conformational change in azurin from Pseudomonas aeruginosa (PaAz). PaAz was characterized extensively with a variety of experimental methods. Experimentally determined pKa values and pH-dependent reduction potentials are used to validate our computational model. It is well known from experiment that the reduction of the copper center is coupled to the protonation of at least two titratable residues (His-35 and His-83) and to the flip of the peptide bond between Pro-36 and Gly-37. Free energy measures of cooperativity are used for a detailed analysis of the coupling between protonation, reduction and conformational change in PaAz. The reduction of the copper center, the protonation of His-35 and peptide flip are shown to be cooperative. Our results show that cooperativity free energies are useful in detecting and quantifying thermodynamic coupling between events in biomolecular systems. The protonation of His-35 and the peptide flip are found to be so tightly coupled that these events happen effectively concerted. This concerted change results in a marked alteration of the electrostatic surface potential of azurin that might affect the interaction of azurin with its binding partners.
    BibTeX entry:
    @Article{Ullmann2011PaAzCoop,
      author   = "R. Thomas Ullmann and G. Matthias Ullmann",
      title    = "{Coupling of protonation, reduction and conformational
                   change in azurin from Pseudomonas aeruginosa
                   investigated with free energy measures of cooperativity}",
      journal  = jpcB,
      volume   = "115",
      pages    = "10346-10359",
      year     = 2011,
      doi      = {10.1021/jp204644h},
      abstract = "{We used free energy calculations within a continuum electrostatics
                   model to analyze the coupling of protonation, reduction and
                   conformational change in azurin from Pseudomonas aeruginosa (PaAz).
                   PaAz was characterized extensively with a
                   variety of experimental methods.
                   Experimentally determined $\mathrm{p}K_{\mathrm{a}}$
                   values and pH-dependent reduction potentials are
                   used to validate our computational model.
                   It is well known from experiment that the reduction of
                   the copper center is coupled to the protonation of at least two
                   titratable residues (His-35 and His-83) and to the flip
                   of the peptide bond between Pro-36 and Gly-37.
                   Free energy measures of cooperativity are used for a
                   detailed analysis of the coupling between protonation,
                   reduction and conformational change in PaAz.
                   The reduction of the copper center, the protonation
                   of His-35 and peptide flip are shown to be cooperative.
                   Our results show that cooperativity free energies are
                   useful in detecting and quantifying thermodynamic
                   coupling between events in biomolecular systems.
                   The protonation of His-35 and the peptide flip are
                   found to be so tightly coupled that
                   these events happen effectively concerted.
                   This concerted change results in a marked alteration
                   of the electrostatic surface potential of azurin
                   that might affect the
                   interaction of azurin with its binding partners.
                   }",
    }
    
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    2010

  • R. Thomas Ullmann, G. Matthias Ullmann; 2010; A generalized free energy perturbation theory accounting for end states with differing configuration space volume; J. Phys. Chem. B; vol. 115; p. 507-521; doi 10.1021/jp1093838
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    Abstract: We present a generalized free energy perturbation theory that is inspired by Monte Carlo techniques and based on a microstate description of a transformation between two states of a physical system. It is shown that the present free energy perturbation theory stated by the Zwanzig equation follows as a special case of our theory. Our method uses a stochastic mapping of the end states that associates a given microstate from one ensemble with a microstate from the adjacent ensemble according to a probability distribution. In contrast, previous free energy perturbation methods use a static, deterministic mapping that associates fixed pairs of microstates from the two ensembles. The advantages of our approach are that end states of differing configuration space volume can be treated easily also in case of discrete configuration spaces and that the method does not require the potentially cumbersome search for an optimal deterministic mapping. The application of our theory is illustrated by some example problems. We discuss practical applications for which our findings could be relevant and point out perspectives for further development of the free energy perturbation theory.
    BibTeX entry:
    @Article{Ullmann2010FEP,
      author   = "R. Thomas Ullmann and G. Matthias Ullmann",
      title    = "{A generalized free energy perturbation theory accounting for
                   end states with differing configuration space volume}",
      journal  = jpcb,
      volume   = "115",
      pages    = "507-521",
      year     = 2010,
      doi      = {10.1021/jp1093838},
      abstract = "{We present a generalized free energy perturbation theory
                   that is inspired by Monte Carlo techniques and based
                   on a microstate description of a transformation between
                   two states of a physical system.
                   It is shown that the present free energy perturbation theory
                   stated by the Zwanzig equation follows as a special case of
                   our theory.
                   Our method uses a stochastic mapping of the end states
                   that associates a given microstate from one ensemble with a microstate
                   from the adjacent ensemble according to a probability distribution.
                   In contrast, previous free energy perturbation methods use a static,
                   deterministic mapping that associates fixed pairs of microstates from
                   the two ensembles.
                   The advantages of our approach are that end states of differing configuration
                   space volume can be treated easily also in case of discrete configuration spaces
                   and that the method does not require the potentially cumbersome search for an
                   optimal deterministic mapping.
                   The application of our theory is illustrated by some example problems.
                   We discuss practical applications for which our findings could
                   be relevant
                   and point out perspectives for further development
                   of the free energy perturbation theory.
                   }",
    }
    
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    2007

  • Torsten Becker, R. Thomas Ullmann, G. Matthias Ullmann; 2007; Simulation of the electron transfer between the tetraheme subunit and the special pair of the photosynthetic reaction center using a microstate description; J. Phys. Chem. B; vol. 111; p. 2957-2968; doi 10.1021/jp066264a
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    Abstract: Charge transfer through biological macromolecules is essential for many biological processes such as, for instance, photosynthesis and respiration. Protons or electrons are transferred between titratable residues or redox-active cofactors, respectively. Transfer rates between these sites depend on the current charge configuration of neighboring sites. Here, we formulate the kinetics of charge-transfer systems in a microstate formalism. A unique transfer rate constant can be assigned to the interconversion of microstates. Mutual interactions between sites participating in the transfer reactions are naturally taken into account. The formalism is applied to the kinetics of electron transfer in the tetraheme subunit and the special pair of the reaction center of Blastochloris viridis. It is shown that continuum electrostatic calculations can be used in combination with an existing empirical rate law to obtain electron-transfer rate constants. The re-reduction kinetics of the photo-oxidized special pair simulated in a microstate formalism is shown to be in good agreement with experimental data. A flux analysis is used to follow the individual electron-transfer steps.
    BibTeX entry:
    
    @Article{Becker2007,
      author   = "Torsten Becker and R. Thomas Ullmann and G. Matthias Ullmann",
      title    = "{Simulation of the electron transfer between the tetraheme subunit
                   and the special pair of the photosynthetic reaction center using
                   a microstate description}",
      journal  = jpcb,
      volume   = "111",
      pages    = "2957-2968",
      year     =  2007,
      doi      = "10.1021/jp066264a",
      abstract = "{Charge transfer through biological macromolecules is essential
                   for many biological processes such as, for instance, photosynthesis
                   and respiration. Protons or electrons are transferred between
                   titratable residues or redox-active cofactors, respectively.
                   Transfer rates between these sites depend on the current charge
                   configuration of neighboring sites. Here, we formulate the kinetics
                   of charge-transfer systems in a microstate formalism. A unique
                   transfer rate constant can be assigned to the interconversion
                   of microstates. Mutual interactions between sites participating
                   in the transfer reactions are naturally taken into account.
                   The formalism is applied to the kinetics of electron transfer
                   in the tetraheme subunit and the special pair of the reaction
                   center of Blastochloris viridis. It is shown that continuum
                   electrostatic calculations can be used in combination with an
                   existing empirical rate law to obtain electron-transfer rate
                   constants. The re-reduction kinetics of the photo-oxidized
                   special pair simulated in a microstate formalism is shown to
                   be in good agreement with experimental data. A flux analysis
                   is used to follow the individual electron-transfer steps.
                   }",
    }
    
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