Reactivity Tracking of an Enzyme Progress Coordinate

The reactivity of individual solvent-coupled protein configurations is used to track and resolve the progress coordinate for the core reaction sequence of substrate radical rearrangement and hydrogen atom transfer in the ethanolamine ammonia-lyase (EAL) enzyme from Salmonella enterica. The first-order decay of the substrate radical intermediate is the monitored reaction. Heterogeneous confinement from sucrose hydrates in the mesophase solvent surrounding the cryotrapped protein introduces distributed kinetics in the non-native decay of the substrate radical pair capture substate, which arise from an ensemble of configurational microstates. Reaction rates increase by >103-fold across the distribution to approach that for the native enabled substate for radical rearrangement, which reacts with monotonic kinetics. The native progress coordinate thus involves a collapse of the configuration space to generate optimized reactivity. Reactivity tracking reveals fundamental features of solvent–protein-reaction configurational coupling and leads to a model that refines the ensemble paradigm of enzyme catalysis for strongly adiabatic chemical steps.

Figures S10 Figure S1. EPR spectra of the cob(II)alamin-substrate radical pair in EAL. Figure S2. Cob(II)alamin in EAL under anaerobic condition with 0, 2, 4, and 5 % (w/v) sucrose. Figure S3. Representative multi-microstate model and dependence of the observed decay rate constants and amplitudes on microstate interconversion rate. Figure S4. Observed first-order rate constants and power law distribution of rate constants for different components of the 1 H-substrate radical decay in EAL samples cryotrapped in the presence of the proteins, bovine serum albumin and lysozyme.
Tables S14 Table S1. Kinetic parameters for biexponential and power law function fits to the observed 1 H-substrate radical decay reactions at 0, 2, 4, and 5 % sucrose. Table S2. Kinetic parameters for biexponential and power law function fits to the observed 2 H-substrate radical decay reactions at 0, 2, 4, and 5 % sucrose. Table S3. Sucrose concentration dependence of the 1 H/ 2 H isotope effect on the observed substrate radical decay rate constants. Table S4. Kinetic parameters for biexponential and power law function fits to the observed 1 H-substrate radical decay reactions in the presence of bovine serum albumin (BSA) and lysozyme (LYZ), and for 2% sucrose control.
The power law dependence implies a distribution of monoexponential rate constants, !"#,#,) , for decay from states, i. 4 The probability, ) , of state i with associated decay rate constant, !"#,#,) , is given by: where Γ(n) is the gamma function. The mean value of kobs,s,i is given by:

Supporting Text
Multi-state kinetics and calculation of mean observed decay rate constants.
In a system with finite number of connected substates, in which each substate has a potential decay pathway, the time dependence of the concentration of each intermediate, , is described as Eq. S1 where )+ and +) are the rate constants for interconversion between intermediates and , and ) is the intrinsic rate constant characterizing the decay reaction from intermediate .
Eq. S1 can be rewritten as in which ⃗ is a column vector that represents intermediate concentrations, as a function of time, and the transition matrix O is defined as

S7
Since the cryo-trapped substrate radical in the ⦁ state proceeds into product without replenishment between 190 K and 250 K 3 , the system is in a non-steady state, and the observed total population decay ( ⃗ !"# ) can be written as a linear combination of timeindependent eigenvectors ⃗ ) and their time-evolving part exp[−λ ) ].
The −λ ) are the eigenvalues of transition matrix O . Because these intermediates are degenerate substates along the reaction coordinate, all the eigenvalues are assumed to be unique. The α 0 are the linear coefficients, which are explicitly obtained from the expression for the initial conditions, Furthermore, the experimental signal resolved from EPR spectroscopy is the total amplitude from all substrate radical intermediates:

S8
The observation of experimental power-law decay of total amplitude 1!1 ( ), therefore represents a linear combination of eigen-phases (Eq. S7), with the observed amplitude !"#,) = α ) sum( ⃗ ) ) and the observed rate constants !"#,) = λ ) . Namely, To build a simulation with n potential intermediates, a × transition matrix O is generated. Here we assume that the interconversion rate constants are uniform, Eq. S9 is an approximation, that is helpful to simplify the model and to better illustrate the  increasing , all observed rates become faster, and the spread between these rates narrows. When > 10 '2 s -1 , the interconversion rate is more rapid than decay rates, and only one averaged exponential decay is observed, ⟨ ) ⟩, that lies within the envelope of the ) . This is the condition for substrate radical decay from the ⦁ state in EAL in the absence of sucrose (0% condition), for which the decay is monoexponential (single rate constant, !"#,# ) (Figure 2 C, D).
The simple multi-state kinetic model considered here supports the interpretation of the monoexponential and power-law decay kinetics of the ⦁ state in EAL as caused by the limits of restricted interconversion (2-5 % sucrose) and rapid interconversion (0% sucrose).    Table S4. Table S1. Kinetic parameters for biexponential and power law function fits to the observed 1 H-substrate radical decay reactions at 0, 2, 4, and 5 % sucrose.   * In the absence of the H-isotope effect, TU,0 *T >> VW,0 *T , WV,0 *T , and VW,0 *T is rate-determining for the substrate radical decay reaction. Therefore, 89:,0 *T = VW,0 *T 5 .

Figures
** The condition, 89:,0 *T = VW,0 *T , entails X,0 *T <<1, and therefore, negligible. † The 1 H/ 2 H-isotope effect on the RR step is assumed to be negligible 5 . Under this assumption, the values of VW,0 *T and VW,0 ,T are equal, and the measured value of VW,0 *T is used for VW,0 ,T . only. The 2 H condition is assumed to influence the barrier for HT, or kHT, only 5 .