727 research outputs found
A mathematical view on the decoupled sites representation
No full textThe decoupled sites representation (DSR) is a theoretical instrument which allows to regard complex pH titration curves of biomolecules with several interacting proton binding sites as composition of isolated, non-interacting sites, each with a standard Henderson-Hasselbalch titration curve. In this work, we present the mathematical framework in which the DSR is embedded and give mathematical proofs for several statements in the periphery of the DSR. These proofs also identify exceptions. To apply the DSR to any molecule, it is necessary to extend the set of binding energies from to a stripe within . An important observation in this context is that even positive interaction energies (repulsion) between the binding sites will not guarantee real binding energies in the decoupled system, at least if the molecule has more than four proton binding sites. Moreover, we show that for a given overall titration curve it is not only possible to find a corresponding system with an interaction energy of zero but with any arbitrary fix interaction energy. This result also effects practical work as it shows that for any given titration curve, there is an infinite number of corresponding hypothetical molecules. Furthermore, this implies that-using a common definition of cooperative binding on the level of interaction energies-a meaningful measure of cooperativity between the binding sites cannot be defined solely on the basis of the overall titration. Consequently, all measures of cooperativity based on the overall binding curve do not measure the type of cooperativity commonly defined on the basis of interaction energies. Understanding the DSR mathematically provides the basis of transferring the DSR to biomolecules with different types of interacting ligands, such as protons and electrons, which play an important role within electron transport chains like in photosynthesis.Deutsche Forschungsgemeinschaft [SFB 840
On the interaction of different types of ligands binding to the same molecule Part II: Systems with n to 2 and n to 3 binding sites
No full textIn the first part of this work we formulated the decoupled sites representation for two different types of ligands and highlighted special properties of the case of n binding sites for ligand L (1) and one binding site for ligand L (2). Moreover, for this case, we identified the microstate constants as unique components all decoupled molecules share. In the second part on hand, we investigate the cases with (n, 2) and (n, 3) binding sites. As it is difficult to solve the system of equations occurring when a molecule with more than one binding site for both ligands shall be decoupled, we present applicable calculation methods which exploit the special structure of the system of equations. Moreover, we investigate which unique properties all decoupled molecules share and show that for two different decoupled molecules with the same binding polynomial, not all microstate constants of a certain macrostate are permutations of the microstate constants of the other molecule
On the interaction of different types of ligands binding to the same molecule Part I: the transfer of the decoupled sites representation
No full textThe decoupled sites representation (DSR) for one type of ligand allows to regard complex overall titration curves as sum of classical Henderson-Hasselbalch (HH) titration curves. In this work we transfer this theoretical approach to molecules with different types of interacting ligands (e.g. protons and electrons), prove the existence of decoupled systems for n (1) and one binding sites for two different ligands, and point out some difficulties and limits of this transfer. A major difference to the DSR for one type of ligand is the loss of uniqueness of the decoupled system. However, all decoupled systems share a unique set of microstate probabilities and each decoupled system corresponds to a certain permutation of these microstate probabilities. Moreover, we show that the titration curve of a certain binding site in the original system can be regarded as linear combination of the titration curves of the individual sites of the decoupled system if the weights of the linear combination are substituted by functions in the activity of the second ligand. In the underlying model with only pairwise interaction, an important observation of our theoretical investigation is the following: Even though the binding sites of ligand L (1) may not interact directly, they can show secondary interaction due to the interaction with the second type of ligand. This means, if the activity of the second ligand is fixed and we regard the 1-dimensional titration curve of an individual binding site for ligand L (1) depending on its activity, we may observe a strong deviation from the classical HH shape in spite of non-interacting sites for ligand L (1)
The Meaning of the Decoupled Sites Representation in Terms of Statistical Mechanics and Stochastics
No full textThe investigation of thermodynamic properties of ligand binding is a classical field of (bio)chemistry and (bio)physics. Commonly, an algebraic description using polynomials (e.g. the binding polynomial) and rational functions (e.g. titration curves) is used to characterize systems of molecules and their ligand(s). However, the algebraic model is a result of the probabilistic setup of statistical mechanics and its concept of the Grand Canonical Partition Function. In this work, we reconsider the decoupled sites representation (DSR), a theoretical tool to regard an overall titration curve as sum of classical Henderson-Hasselbalch ligand binding curves from a stochastic point of view. Our work closes the circle from the initial stochastic model, to an algebraic description in which the DSR was developed and analyzed, back to its meaning in statistical mechanics and stochastics. In this regard, we translate results in the periphery of the DSR which were derived within the algebraic model into stochastics. The shifted point of view facilitates some proofs and physical interpretations and provides the basis for future work which might investigate how certain "phenomenona of the algebraic concept can be interpreted stochastically.DFG GRK [1640
The Meaning of the Decoupled Sites Representation in Terms of Statistical Mechanics and Stochastics
No full textThe investigation of thermodynamic properties of ligand binding is a classical field of (bio)chemistry and (bio)physics. Commonly, an algebraic description using polynomials (e.g. the binding polynomial) and rational functions (e.g. titration curves) is used to characterize systems of molecules and their ligand(s). However, the algebraic model is a result of the probabilistic setup of statistical mechanics and its concept of the Grand Canonical Partition Function. In this work, we reconsider the decoupled sites representation (DSR), a theoretical tool to regard an overall titration curve as sum of classical Henderson-Hasselbalch ligand binding curves from a stochastic point of view. Our work closes the circle from the initial stochastic model, to an algebraic description in which the DSR was developed and analyzed, back to its meaning in statistical mechanics and stochastics. In this regard, we translate results in the periphery of the DSR which were derived within the algebraic model into stochastics. The shifted point of view facilitates some proofs and physical interpretations and provides the basis for future work which might investigate how certain "phenomenona of the algebraic concept can be interpreted stochastically.DFG GRK [1640
Structure of the non-redox-active tungsten/[4Fe:4S] enzyme acetylene hydratase
No full textThe tungsten iron sulfur enzyme acetylene hydratase stands out from its class because it catalyzes a nonredox reaction, the hydration of acetylene to acetaldehyde. Sequence comparisons group the protein into the dimethyl sulfoxide reductase family, and it contains a bis-molybdopterin guanine dinucleotide-ligated tungsten atom and a cubane-type [4Fe:4S] cluster. The crystal structure of acetylene hydratase at 1.26 Å now shows that the tungsten center binds a water molecule that is activated by an adjacent aspartate residue, enabling it to attack acetylene bound in a distinct, hydrophobic pocket. This mechanism requires a strong shift of pKa of the aspartate, caused by a nearby low-potential [4Fe:4S] cluster. To access this previously unrecognized W Asp active site, the protein evolved a new substrate channel distant from where it is found in other molybdenum and tungsten enzymes.publishe
Photoswitching of the Fluorescent Protein asFP595: Mechanism, Proton Pathways, and Absorption Spectra
No full textMolecular light‐switch: Off–on switching of the fluorescence of the protein asFP595 involves a trans–cis isomerization. Mixed quantum/classical simulations elucidate the spectroscopic properties of asFP595 and give detailed insights into the photoswitching mechanism. The conformational trans–cis switching triggers a proton‐transfer cascade between the chromophore and adjacent amino acids
On the Complexity of Finding a Sparse Connected Spanning Subgraph in a Non-Uniform Failure Model
No full textWe study a generalization of the classic Spanning Tree problem that allows for a non-uniform failure model. More precisely, edges are either safe or unsafe and we assume that failures only affect unsafe edges. In Unweighted Flexible Graph Connectivity we are given an undirected graph G = (V,E) in which the edge set E is partitioned into a set S of safe edges and a set U of unsafe edges and the task is to find a set T of at most k edges such that T -{u} is connected and spans V for any unsafe edge u ∈ T. Unweighted Flexible Graph Connectivity generalizes both Spanning Tree and Hamiltonian Cycle. We study Unweighted Flexible Graph Connectivity in terms of fixed-parameter tractability (FPT). We show an almost complete dichotomy on which parameters lead to fixed-parameter tractability and which lead to hardness. To this end, we obtain FPT-time algorithms with respect to the vertex deletion distance to cluster graphs and with respect to the treewidth. By exploiting the close relationship to Hamiltonian Cycle, we show that FPT-time algorithms for many smaller parameters are unlikely under standard parameterized complexity assumptions. Regarding problem-specific parameters, we observe that Unweighted Flexible Graph Connectivity admits an FPT-time algorithm when parameterized by the number of unsafe edges. Furthermore, we investigate a below-upper-bound parameter for the number of edges of a solution. We show that this parameter also leads to an FPT-time algorithm.Discrete Mathematics and Optimizatio
Enzymatic Birch reduction via hydrogen atom transfer at [4Fe-4S]-OH2 and [8Fe-9S] clusters
No full text11 p.-7 fig.The alkali metal- and ammonia-dependent Birch reduction is the classical synthetic method for achieving dihydro additions to arenes, typically yielding 1,4-cyclodienes. A mild biological alternative to this process are 1,5-dienoyl-coenzyme A (CoA)-forming class I and II benzoyl-CoA reductases (BCRs), widely abundant key enzymes in the biodegradation of aromatic compounds at anoxic environments. To obtain a comprehensive mechanistic understanding of class I BCR catalysis, we produced the active site subunits from a denitrifying bacterium and determined the X-ray structure of its substrate and product complexes at 1.4 Å revealing non-canonical double-cubane [8Fe-9S] and active site aqua-[4Fe-4S] clusters. Together with kinetic, spectroscopic and QM/MM studies, we provide evidence for a radical mechanism with a [4Fe-4S] cluster-bound water molecule acting as hydrogen atom and electron donor at potentials beyond the biological redox window. An analogous Birch-like radical mechanism is applied by class II BCRs with the catalytic water bound to a tungsten-bis-metallopterin cofactor. The use of activated, metal-bound water ligands as hydrogen atom donor serves as a basic blueprint for future enzymatic or biomimetic Birch reduction processes.This research was supported by the German Research council within RTG 1976, grant number 235777276 (to J.F. and M.B.) and SPP 1927: Iron-Sulfur for Life - Project number 311144407 (to G.M.U.). The EPR spectrometer upgrade and closed-cycle cryostat (A.J.P.) was funded by the DFG (248/320-1, project number 444947649) and the government of Rhineland-Palatinate. We thank Hartmut Michel for continuous support and the staff at the PXII beamline at the Swiss-Light source, Villigen, Switzerland for help during data collection. The research stay of U.F.A. at the University of Freiburg was funded by the grant PID2022-142540OB-I00 from the Spanish Ministry of Science, Innovation and Universities.Peer reviewe
Robuste Finite Elemente Löser für molekulare elektrostatische Energie Berechnungen
No full textACKNOWLEDGEMENTS PREAMBLE ABBREVIATIONS I. THEORY INTRODUCTION CLASSICAL
ELECTROSTATICS IN MOLECULAR SYSTEMS Coulomb’s Law Gauss’s Law and the Poisson
Equation The Debye-Hückel Approximation The Poisson-Boltzmann Equation
DISCRETIZING THE LINEARIZED POISSON-BOLTZMANN EQUATION The Finite Difference
Method The Boundary Element Method The Finite Element Method SOLVING THE
LINEAR EQUATION SYSTEM Direct Methods Iterative Methods Multigrid Method
Multifrontal Method DISCRETIZATION PITFALLS Artificial Grid Energy Surface
Discretization Pitfalls APPLICATIONS Computation of the Electrostatic
Solvation Energy Determination of pKA Values in Proteins II. MFES: A ROBUST
MOLECULAR FINITE ELEMENT SOLVER FOR ELECTROSTATIC ENERGY COMPUTATIONS SUMMARY
AND DISCUSSION PEER-REVIEWED PAPER III. PKA IN PROTEINS SOLVING THE POISSON-
BOLTZMANN EQUATION WITH FINITE ELEMENTS IV. MFES IMPLEMENTATION AND MFES+ WEB
SERVICES CONCLUSION AND OUTLOOK SUMMARY BIBLIOGRAPHY LIST OF FIGURES LIST OF
TABLES LIST OF PUBLICATIONS APPENDIX A UNITS AND CONVERSIONS B BORN MODEL
DERIVATION C SUPPORTING INFORMATION TO CHAPTER II D SUPPORTING INFORMATION TO
CHAPTER IIIIn this interdisciplinary work, the computer program mFES (molecular Finite
Element Solver) has been developed for solving electrostatic problems
associated with proteins. It is fast and yields results with a higher accuracy
than that obtained with the more traditional software based on finite
difference methods. Electrostatics is an important topic in computational
chemistry. This work focuses on the computation of electrostatic properties of
small molecules like peptides as well as large proteins like viruses. mFES is
based on the well-defined Finite Element (FE) method and solves the linear
Poisson-Boltzmann equation, a second-order partial differential equation,
where a solution algorithm was for instance described by Warwicker and Watson
in 1982 [129]. Besides electrostatic potentials Φ(r), single and double energy
differences ∆G and ∆∆G are computed to determine different properties, like
the electrostatic solvation energy and pKA values, using a robust generation
of the molecular model. Fundamental progress has been made in this thesis
which continues preceding approaches of Friesner et al. [130]–[132] and Holst
et al. [41], [133] mostly more than a decade ago. mFES is based on LSMS [2]
and NETGEN [1] in building molecular models and on MUMPS [42]–[44] in solving
the linear Poisson-Boltzmann equation. The key improvement lies in the way
molecular surfaces (solvent excluded surfaces) are generated. While the
molecular surface itself is well-defined [54], [134], [135], it remained
difficult up to now to compute very precise molecular surfaces without
singularities. Using the advancing front method [51], [52] implemented by
Schöberl et al. [1], this problem is solved by producing a triangulation of
the molecular surface which is controllable by parameters like the average
edge length of the triangles on the molecular surface. Another key improvement
of the Finite Element solver lies in the way the tetrahedral volume elements
are optimized, thus producing an overall molecular model of high quality.
Comparing the FE method with Boundary Element and Finite Difference methods,
the focus lies on the latter because it is well-established and most commonly
used in the electrostatics community. The Finite Element methods that have
been available up to now and which are directly compared are not able to yield
an accuracy which could compete with that of the Finite Difference method.
This is now solved by mFES. Electrostatic solvation energies are computed for
average- and large-sized proteins (bovine pancreatic trypsin inhibitor,
barnase, lysozyme, cyctochrome c oxidase) and the adenovirus serves as an
example for a very large protein with nearly 193,000 atoms, including hydrogen
atoms. Improvements in accuracy resulted in electrostatic solvation energy
differences as high as 30 [kJ/mol] for average-sized proteins. All molecular
systems based on the FE method needed much fewer equations to solve the
Poisson-Boltzmann equation compared with using the FD method and still higher
accuracy is achieved. A proof of concept is the computation of pKA values with
mFES. pKA values for lysozyme and other proteins are computed and compared
with 342 experimentally measured pKA values as well as with results obtained
with the Finite Difference method. These computations are done using the
classical Karlsberg+ program developed by Kieseritzky et al. [102], [103].
pKA-RMSD values with respect to measured values computed by mFES are of the
same quality compared with the values obtained by the FD method. mFES+ web
application services for using mFES without the need for a local installation
are available at: http://agknapp.chemie.fu-berlin.de/mfes . Installation
routines for the stand-alone mFES program are provided as well as example runs
and a documentation to facilitate the use of mFES in other frameworks.In dieser interdisziplinären Arbeit ist das Softwareprogramm mFES (molecular
Finite Element Solver) entstanden, um elektrostatische Fragestellungen für
Proteine zu lösen. mFES ist schnell und berechnet Ergebnisse mit höherer
Präzision als traditionelle Software, die auf der Finite Differenzen Methode
basiert. Die Elektrostatik ist ein wichtiges Thema der computergestützten
Chemie. Diese Arbeit legt ihren Fokus auf die Berechnung elektrostatischer
Eigenschaften kleiner Moleküle wie Peptide bis hin zu großen Proteine wie z.B.
Viren. mFES basiert auf einer wohldefinierten Finite Elemente (FE) Methode und
löst die lineare Poisson-Boltzmann-Gleichung, eine partielle
Differentialgleichung zweiter Ordnung, für die zum Beispiel Warwicker und
Watson 1982 einen Lösungsalgorithmus beschrieben haben [129]. Neben
elektrostatischen Potentialen Φ(r) werden einfache und doppelte
Energiedifferenzen, ∆G und ∆∆G, berechnet, um verschiedene Eigenschaften, wie
elektrostatische Solvatisierungsenergien und pKA Werte, zu berechnen, wofür
robuste molekulare Modelle generiert werden. Fundamentale Fortschritte
gelangen mit dieser Arbeit, welche an die Arbeiten von u. a. Friesner et al.
[130]–[132] und Holst et al. [41], [133] anknüpft, die meist mehr als ein
Jahrzehnt zurückliegen. mFES benutzt zur Generierung molekularer Modelle LSMS
[2] und NETGEN [1] und zur Lösung der linearen Poisson-Boltzmann Gleichung
MUMPS [42]–[44]. Eine entscheidende Verbesserung der elektrostatischen
Berechnungen liegt in der Art der Erzeugung molekularer Oberflächen (solvent
excluded surface). Obwohl diese Oberfläche wohldefiniert ist [54], [134],
[135], ist es bis heute schwierig, eine nahezu exakte Oberfläche ohne
Singularitäten zu generieren. Durch die Nutzung der Advancing-Front-Methode
[51], [52], welche u. a. von Joachim Schöberl [1] entwickelt wurde, wird
dieses Problem gelöst, indem eine Triangulation der molekularen Oberfläche
generiert wird, die durch verschiedene Parameter, beispielsweise die mittlere
Kantenlänge der Dreiecke, kontrollierbar ist. Eine weitere entscheidende
Verbesserung des Finite Elemente Lösers liegt in der Art der Optimierung der
Tetraeder-Volumenelemente, welche ein molekulares Gesamtmodell mit hoher
Qualität erzeugt. Beim Vergleich der Finiten Elemente Methode mit der Rand
Elemente Methode und der Finite Differenzen Methode liegt der Fokus auf
letzterer, weil diese etabliert ist und überwiegend von den Experten benutzt
wird. Die bisher verfügbaren Finite Elemente Methoden, die direkt miteinander
verglichen werden, liefern im Vergleich zur Finite Differenzen Methode keine
hohe Genauigkeit. Dieses Problem wird mit mFES gelöst. Elektrostatische
Solvatisierungsenergien wurden für kleine und große Proteine berechnet
(Rinderpankreas-Trypsininhibitor, Barnase, Lysozym und Cyctochrom-c-Oxidase),
aber auch für Adenovirus-Proteine, welche ein Beispiel für sehr große Proteine
mit fast 193.000 Atomen inklusive der Wasserstoffatome darstellen.
Verbesserungen der Präzision ergaben elektrostatische Solvatisierungsenergie
Unterschiede von bis zu 30 [kJ/mol] für durchschnittlich große Proteine. Alle
molekularen Systeme, die auf der Finiten Elemente Methode basieren, benötigen
hierbei weniger lineare Gleichungen zur Lösung der Poisson-Boltzmann-Gleichung
als die Finite Differenzen Methode. Ein Machbarkeitsbeweis ist die Berechnung
von pKA-Werten mit mFES. pKA-Werte wurden für Lysozym und andere Proteine
berechnet und mit 342 experimentell ermittelten pKA-Werten und mit den
Ergebnissen der Finite-Differenzen-Methode verglichen. Für die Berechnungen
wurde das klassische Programm Karlsberg+ benutzt, welches von Kieseritzky et
al. [102], [103] entwickelt wurde. pKA-RMSD-Werte in Bezug auf gemessene Werte
beweisen, dass die mit mFES berechneten pKA-Werte die gleiche Güte haben wie
pKA-Werte, die mit der FD-Methode berechnet wurden. Webservices für die
Nutzung von mFES+ web ohne eine lokale Installation sind verfügbar unter:
http://agknapp.chemie.fu-berlin.de/mfes . Routinen zur Installation von mFES
stehen zur Verfügung, ebenso einige Beispielrechnungen und eine Dokumentation,
um die Möglichkeit bereitzustellen, mFES in andere Frameworks einzubinden
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