K eq = k2 f / k2 r = C / B, 1. dA(t)/dt = (T/(1+ K eq )) k1 r – A k1 f 2. dT(t)/dt = A k1 f – (T/(1+ K eq )) k1 r – T K eq /(1+ K eq )k3 f 3. dE(t)/dt = T K eq /(1+ K eq ) k3 f Fig. 2. HJ resolution assay. Resolution reaction starting with HJ (B & C) as substrates. We have isolated the HJ intermediates, re-load them with proteins, and monitor the second round of catalytic events or the reverse of the first round of catalytic events attL attP TS2BS1 NC This project has been funded by NSF grant Interdisciplinary Training in Biology and Mathematics to Dr. A. M. Segall and Dr. P. Salamon, and by NIH grant R01-GM52847 to Dr. A. M. Segall. TS junction isomer is biased toward reverse resolution. BS junction isomer is biased toward forward resolution. WT junction isomer resolves to attP/attB (E) greater than attL/attR (A), which is similar to BS junction isomer. Investigating Site-Specific Recombination of Bacteriophage Lambda Using Biochemical and Mathematical Approaches Steven Esquivel, Yi-An Lai, Ryan Blanchet, Kenny Sokolowski, Peter Salamon‡, Anca Segall‡ - San Diego State University, San Diego, California WT1 NC Fig. 3. Quantification of HJ resolution assay (Fig 2). The quantification data was fitted to models that solve for the forward (dE/dt) and reverse rate (dA/dt). B A EC Determine the rate constants of the steps in phage lambda site-specific recombination. NC NC Fig. 4. Three dimension differential equation system used to represent the mechanism steps of the recombination reaction. Modeling The Four Dimension Differential Equation System HJ attP attL attB Table 1. Ratios of E vs. A for the three different species of Holliday Junction. The table includes the proportions (dE/dA) of different HJ and the comparison of their relative equilibrium constants with one another. Compare experimental data with another mathematical approach using Matlab. From this, we should be able to run computational simulations of this event using four differential equations A(t), B(t), C(t) and E(t). From this, we can incorporate actual values at specific time points and solve the differential equations to find the relative equilibrium. Optimize conditions for the separation of the different Holliday junction isomers complexed with proteins, in order to simplify and to extend our analysis. Fig. 5. Simulation of the four dimension differential equation system solved using the Matlab ode23 command. We are utilizing a four dimension differential equation system by using a four by one column vector dydt(1,1)  dydt(4,1), where (1,1) represents the first row and first column of a matrix we call dydt. The matrix dydt is thus a column vector (4 x 1) – (rows x columns) that consists of the 4 differential equations dA/dt  dE/dt. ‡ Co-Senior Author Int IHF Xis + Fis Int IHF Fig. 1. Schematic view of excision recombination. Two DNA substrates (A) recombine resulting in a HJ intermediate (B & C) that may resolve to form products depicted as E. Bacteriophage lambda lytic cycle Bacteriophage lambda integrates and excises its genomic DNA from bacteria cells via site-specific recombination. Integrase (Int), a tyrosine recombinase, catalyses phage λ site-specific recombination (both integration & excision). Accessory proteins: Integration host factor (IHF) bends DNA and excisionase (Xis) stimulates the excision reaction while inhibiting integration. Catalytic event in site-specific recombination Cre R173K/loxP-HJ X-ray crystal structure of Cre complexes with loxP site holliday junction (HJ) with symmetry. Resolution: 2 Å K. Ghosh and G. Van Duyne, Univ. of Pennsylvania Cre protein is a prototypical member in tyrosine recombinase family.