Behringer, H.

Schmid, F.

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John von Neumann Institute for ComputingCorrelation Effects in Protein-ProteinRecognitionH. Behringer, F. Schmidpublished inFrom Computational Biophysics to Systems Biology (CBSB08),Proceedings of the NIC Workshop 2008,Ulrich H. E. Hansmann, Jan H. Meinke, Sandipan Mohanty,Walter Nadler, Olav Zimmermann (Editors),John von Neumann Institute for Computing, Ju¨lich,NIC Series, Vol. 40, ISBN 978-3-9810843-6-8, pp. 165-168, 2008.c© 2008 by John von Neumann Institute for ComputingPermission to make digital or hard copies of portions of this work forpersonal or classroom use is granted provided that the copies are notmade or distributed for profit or commercial advantage and that copiesbear this notice and the full citation on the first page. To copy otherwiserequires prior specific permission by the publisher mentioned above.http://www.fz-juelich.de/nic-series/volume40Correlation Effects in Protein-Protein RecognitionHans Behringer and Friederike SchmidFakulta¨t fu¨r Physik, Universita¨t Bielefeld, D-33615 Bielefeld, GermanyE-mail: {behringer, schmid}@physik.uni-bielefeld.deCorrelation effects in the distribution of hydrophobic and polar residues are investigated withinan idealised coarse-grained model for the recognition of two rigid biomolecules such as pro-teins. To this end, a two-stage approach is adopted where the biomolecules are first optimisedwith respect to each other and afterwards their selectivity is tested in the presence of othermolecules. Correlations lead to different optimum characteristic lengths of the hydrophobicand polar patches for the design of the two biomolecules on the one hand and their selectivityin the presence of other molecules on the other hand.1 IntroductionBiomolecular recognition, that is the ability of a biomolecule to interact specifically withanother molecule in an heterogeneous environment of structurally similar rival molecules,is an essential component in biological systems. The recognition process between twomolecules is governed by a complicated interplay of non-covalent interactions of strengthscomparable to the thermal energy1. This implies that the study of idealised models withmethods from statistical physics might lead to valuable insight into this problem.2 Model and General ApproachIn this work we consider protein-protein recognition on a coarse-grained level in the frame-work of idealised models. The biomolecules are assumed to undergo no refolding duringthe association process which is a justified assumption for most protein-protein recognitionprocesses1. Motivated by the observation that hydrophobicity is the major driving force inmolecular recognition1 we describe the type of the residue at the position i = 1, . . . , N ofthe interface by a binary variable σi ∈ {±1} for the target molecule and by θi ∈ {±1} forthe interaction partner2. We then model the energetics at the two-dimensional interface byH(σ, θ;S) = −εN∑i=11 + Si2σiθi (1)as a direct contact interaction of strength ε. The variable Si takes on the two values ±1and describes the local fit of the shape of the molecules at the interface resulting from arearrangement of the amino acid side chains when the complex is formed1.To study the recognition process between two rigid biomolecules we adopt a two-stageapproach. For a fixed target sequence σ(t) = (σ(t)1 , . . . , σ(t)N ) we first design an ensembleof probe molecules θ at a design temperature 1/βD leading to the distribution P (θ|σ(t)) =1ZD∑S exp(−βDH(σ(t), θ;S)). In a second step the free energy difference of associationat temperature 1/β is calculated for the interaction of the probe ensemble with the targetmolecule σ(t) and a structurally different rival molecule σ(r). In this step the free energy of165the interaction of the molecule σ(α) with a particular probe sequence θ has to be averagedwith respect to the distribution P (θ|σ(t)) giving finally the selectivity ∆F = Ftarget−Frival.A negative ∆F then signals recognition of the target.For the majority of real protein-protein complexes the appearance of extended but fairlysmall patches of residues of the same type has been reported3. Biomolecular binding seemsalso to be strongly influenced by small-scale structures4. We therefore consider moleculeswhich have correlated recognition sites at the interface with extended patches of residuesof the same type. This can be incorporated into our model by adding additional correlationterms likeHcor = −γ∑〈i,j〉θiθj − µ∑iθi. (2)to the Hamiltonian of the system. The correlation parameters γ and µ (for the differ-ent types of molecules) are then used to fix the hydrophobicity and correlation length onthe recognition sites. The average extension of the patches of residues from the sameclass is used as a measure for the correlation length of the finite system. Introducing thecomplementarity parameter K =∑i σiθi which measures the structural fit of the twobiomolecules σ and θ at the interface, the selectivity averaged over all targets and rivalswith the same correlation properties turns out to be the negative difference between theaveraged complementarity with the target and the rival molecules, respectively5.3 ResultsIn this section molecular recognition for target and rival molecules with a fixed average hy-drophobicityh = 0.4 per residue and a fixed (to length unity normalised) correlation lengthλ = 0.263 is considered within the model discussed above. The selectivity is studied as afunction of the correlation length of the recognition site of the probe molecules. Figure 1-0.5 0 0.5 1K00.010.020.030.040.050.06 D(K)-0.5 0 0.5 1K00.010.020.030.040.050.06Figure 1. Distribution of the complementarity of the probe molecules with the target molecules (solid curve) andthe rival molecules (shaded curve) for uncorrelated (left) and correlated probe molecules (with correlation lengthλp = 0.25, right).shows the distributions of the complementarities. For uncorrelated probe molecules thedistribution for the complementarity with the target molecules is clearly separated from166the one with the rival molecules and shifted to larger values. This indicates the overallrecognition ability of the system. A moderate increase of the correlation length on theprobe molecules shifts the distribution to larger values of the complementarity so that anincreased selectivity is expected. The first moments of the distributions are shown in fig-ure 2. The average complementarity of the probe molecules with the target is always largerthan that of the probe molecules with the rival. In the extreme cases where the correlationlength tends to the minimum and maximum possible values the two averages become iden-tical indicating that selectivity is lost as the probe molecules are not structured any morewith respect to a particular molecule. The selectivity as shown in figure 2 has an optimumat a correlation length that is shifted below the value corresponding to the optimum of thecomplementarity with the target molecules. A smaller correlation length implies the ap-pearance of an increased number of smaller patches on the recognition site of the probemolecule and hence an entropic profit for the interaction with the target due to more possi-ble ways to align each other favourably. This effect only influences the contributions fromthe target-probe interactions due to the optimisation during the design step. The rival-probeinteraction is not optimised and hence it is insensitive to a matching of structure elements.00.20.40.60.8[<K>]/N0 0.2 0.4 0.6 0.8 1 λp -0.3-0.2-0.1[∆F]/NFigure 2. Upper part: Complementarity with the target molecules (solid curve) and the rival molecules (dashedcurve) as a function of the (normalised) correlation length of the probe molecules (the fixed correlation length ofthe target and rival molecules is shown by the circle). Lower part: Resulting selectivity of the system.AcknowledgmentsWe thank the Deutsche Forschungsgemeinschaft (SFB 613) for financial support.References1. C. Kleanthous, ed., Protein-Protein Recognition (Oxford University Press, Oxford,2000).2. H. Behringer, A. Degenhard, and F. Schmid, Phys. Rev. Lett. 97, 128101, 2006.3. T. A. Larsen, A. J. Olsen, and D. S. Goodsell, Structure 6, 421, 1998.4. T. Bogner, A. Degenhard, and F. Schmid, Phys. Rev. Lett. 93, 268108, 2004.5. H. Behringer and F. Schmid, in preparation.167

Correlation Effects in Protein-Protein Recognition

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Correlation Effects in Protein-Protein Recognition

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