13.2 Stopped-flow and relaxation techniques

Stopped-Flow and Relaxation Techniques

Stopped-flow and relaxation techniques are the primary tools for studying fast chemical reactions that complete in milliseconds or microseconds. Standard mixing-by-hand methods can't capture kinetics on these timescales, so these specialized approaches fill a critical gap. They let you measure rate constants for rapid processes like enzyme catalysis, protein folding, and ligand binding.

The two methods work differently: stopped-flow rapidly mixes reactants and then monitors the reaction as it proceeds, while relaxation techniques start with a system already at equilibrium and give it a sudden jolt. Both generate time-resolved data you can fit to kinetic models to extract rate constants and propose mechanisms.

Principles of Stopped-Flow Techniques

Stopped-flow works by rapidly mixing two reactant solutions and then abruptly halting the flow so you can watch the reaction unfold. It's particularly useful for reactions with half-lives ranging from about 1 millisecond to several seconds.

Here's how the apparatus works:

1. Drive syringes push precise volumes of two reactant solutions toward a mixing chamber at high speed.
2. The mixing chamber combines the solutions thoroughly in under 1 ms (this dead time sets the lower limit on what you can measure).
3. The mixed solution flows into an observation cell, where a detector monitors the reaction in real time.
4. A stop syringe or physical barrier halts the flow abruptly, so the solution sits still in the observation cell while the reaction continues.
5. A detection system records changes in a spectroscopic signal (absorbance, fluorescence, circular dichroism) as the reaction progresses.

The key limitation is the dead time, which is the delay between mixing and the first reliable measurement. For most instruments this is roughly 0.5–2 ms, so any reaction that completes faster than that can't be captured by stopped-flow alone.

Relaxation Techniques for Reaction Kinetics

Relaxation techniques take a different approach. Instead of mixing reactants, you start with a system that's already at equilibrium and then apply a sudden perturbation. The system is now out of equilibrium under the new conditions, and you monitor how it "relaxes" back to a new equilibrium state.

Common types of perturbation:

• Temperature jump (T-jump): A laser pulse or capacitor discharge heats the sample by several degrees in microseconds. This is the most widely used relaxation method.
• Pressure jump (P-jump): A rapid pressure change (often using a burst diaphragm or piezoelectric transducer) shifts the equilibrium of reactions with a nonzero $\Delta V$.
• Electric field jump (E-jump): A strong electric field pulse perturbs equilibria involving charged or polar species.

Because the perturbation itself happens on a microsecond (or faster) timescale, relaxation techniques can probe reactions with half-lives in the microsecond to millisecond range. That makes them faster than stopped-flow and well-suited for processes like protein conformational changes and rapid ligand binding.

The central quantity you extract is the relaxation time $\tau$, which describes how quickly the system returns to equilibrium. For a simple one-step reaction $A \rightleftharpoons B$, the observed relaxation follows a single exponential decay, and:

$\tau = \frac{1}{k_f + k_r}$

where $k_f$ is the forward rate constant and $k_r$ is the reverse rate constant. Note that $\tau$ depends on both rate constants, not just one. The simplified expression $\tau = 1/k$ only applies when one direction dominates or for an irreversible process. For more complex mechanisms (e.g., $A + B \rightleftharpoons C$), the relationship between $\tau$ and the rate constants also involves equilibrium concentrations, so you typically measure $\tau$ at several different concentrations and fit the resulting dependence.

Stopped-Flow vs. Relaxation Techniques

The two methods are complementary. Stopped-flow is often the first choice when you need to initiate a reaction by combining two solutions. Relaxation techniques become necessary when the reaction is too fast for the stopped-flow dead time, or when you want to study elementary steps within an equilibrium process.

Analysis of Stopped-Flow and Relaxation Data

For stopped-flow data:

1. Record the spectroscopic signal (absorbance, fluorescence, etc.) as a function of time after mixing.
2. Plot signal vs. time. Most fast reactions produce an exponential curve. Fit the trace to a single exponential ($S(t) = A \cdot e^{-k_{obs} \cdot t} + C$) or, if the reaction has multiple phases, a multi-exponential model.
3. Extract the observed rate constant $k_{obs}$ from the fit. For a pseudo-first-order reaction, plot $k_{obs}$ against the concentration of the excess reagent. The slope gives the second-order rate constant, and the y-intercept gives the reverse rate constant.

For relaxation data:

1. Record the signal vs. time after the perturbation (e.g., after the T-jump pulse).
2. Fit the relaxation trace to an exponential decay to obtain the relaxation time $\tau$.
3. Calculate the sum of rate constants from $k_f + k_r = 1/\tau$. For a bimolecular equilibrium like $A + B \rightleftharpoons C$, the relationship becomes $1/\tau = k_f([A]_{eq} + [B]_{eq}) + k_r$, so you need to measure $\tau$ at multiple equilibrium concentrations and plot $1/\tau$ vs. $([A]_{eq} + [B]_{eq})$ to separate $k_f$ and $k_r$.

For both methods:

• Always consider how experimental conditions (temperature, pH, ionic strength, concentration) affect the observed kinetics.
• Compare your extracted rate constants with published literature values as a sanity check.
• Use the concentration dependence of $k_{obs}$ or $1/\tau$ to distinguish between proposed mechanisms. A linear dependence suggests a simple one-step process, while curvature or multiple relaxation times point to a multi-step mechanism.