Abstract Histidine acid phosphatases are characterized by the presence of a conserved RHGXRXP motif. One medically important acid phosphatase is the Prostatic Acid Phosphatase (PAP). PAP has been associated with prostate cancer for a long time and has been used as a marker to stage prostate carcinoma. Yet, there is no clear understanding on the functioning of the enzyme in vivo. This thesis work focuses on the characterization of putative ligands and elucidation of the reaction mechanism of PAP using computational methods. The ligand-enzyme complexes were generated by docking and molecular dynamics simulations. The complexes showed that the conserved arginines of RHGXRXP motif are important for binding the highly negatively charged phosphate group. The complexes also highlighted that the active site aspartate (Asp258) should be neutral in the complex and is involved as a general acid-base in the reaction. The studies support that PAP could dephosphorylate the growth factor receptors EGFR and ErbB-2. The studies also found that the majority of tyrosine phosphorylated peptides from these growth factor receptors could bind to PAP. The affinities were assessed based on theoretical calculations and were further confirmed by experimental measurements in the feasible cases. To clearly understand the mechanism of PAP, quantum mechanical methods were employed. The enzymatic reaction involves two steps. In the first step, the phosphate moiety is transferred from the ligand to the conserved histidine. The calculations on the first step of the reaction involved generating the transition state (TS) structures and estimating the respective barriers. The calculations clearly support that Asp258 becomes neutral by picking up the proton from the monoanionic ligand entering the binding site. The proton from neutral Asp258 is later transferred to the leaving group via a water bridge, restoring the negative state of Asp258. The second step involves the hydrolysis of phosphohistidine enzyme intermediate. Using hybrid quantum mechanics/molecular mechanics calculations, it was found that the Asp258 accepts a proton from the nucleophilic water only after the TS is crossed. This proton is possibly then transferred to the free phosphate while it leaves the binding site, restoring the enzyme to its free state. The study highlights the importance of active site arginines in the binding as well as the stabilization of TS. Further, the analysis of TS structures in both the steps showed an associative mechanism, based on the distance of the nucleophilic and the leaving atoms to the phosphate atom. These distances are much smaller than what has been found in other well studied nonmetallo-phopshatases. Thus, the study finds a novel mechanism of enzymic phosphotransfer in PAP mediated catalysis.

We introduce the formalism of supersymmetric quantum mechanics, including super-symmetry charges,Z2-graded Hilbert spaces, the chirality operator and the Wittenindex. We show that there is a one to one correspondence of fermions and bosons forenergies different than zero, which implies that the Witten index measures the dif-ference of fermions and bosons at the ground state. We argue that the Witten indexis the index of an elliptic operator. Quantization of the supersymmetric non-linearsigma model shows that the Witten index equals the Euler characteristic of the un-derlying Riemannian manifold over which the theory is defined. Finally, the pathintegral representation of the Witten index is applied to derive the Gauss-Bonnettheorem. Apart from this we introduce elementary mathematical background in thesubjects of topological invariance, Riemannian manifolds and index theory / Vi introducucerar formalismen f ̈or supersymmetrisk kvantmekanik, d ̈aribland super-symmetryladdningar,Z2-graderade Hilbertrum, kiralitetsoperatorn och Wittenin-dexet. Vi visar att det r ̊ader en till en-korrespondens mellan fermioner och bosonervid energiniv ̊aer skillda fr ̊an noll, vilket medf ̈or att Wittenindexet m ̈ater skillnadeni antal fermioner och bosoner vid nolltillst ̊andet. Vi argumenterar f ̈or att Wittenin-dexet ̈ar indexet p ̊a en elliptisk operator. Kvantisering av den supersymmetriskaicke-linj ̈ara sigmamodellen visar att Wittenindexet ̈ar Eulerkarakteristiken f ̈or denunderliggande Riemannska m ̊angfald ̈over vilken teorin ̈ar definierad. Slutligenapplicerar vi v ̈agintegralrepresentationen av Wittenindexet f ̈or att h ̈arleda Gauss-Bonnets sats. Ut ̈over detta introduceras ocks ̊a grundl ̈aggande matematisk bakrundi ämnena topologisk invarians, Riemmanska m ̊angfalder och indexteori.