Lamm equation

The goal of a SV experiment is to monitor the rate of movement of the solvent—solution boundary (or, the evolution of the radial concentration gradient dc/dx as a function of time), described by the Lamm equation which takes into account the combined effects of sedimentation and diffusion, as follows ... 

As described in further details in Section 5, we analyze the scans using the software DCDT+ (Philo, 2006), which converts the raw concentration profiles into time derivatives (dc/dt) and fits these values to approximate unbounded solutions of the Lamm equation (Philo, 2000 Stafford, 1994). As the rotor speed (ft)) and the concentration of the macromolecules (c) are known, and the time (t) and the radial concentration distribution [c(x, f)] are obtained from the scans of absorbance profiles, the fitting yields values of s and D. As both parameters are dependent on the solvent viscosity and temperature, they are transformed to standard values with reference to a standard temperature (20 °C) and a standard solvent (water) and reported as 52o,w and /92o,w This standardization allows analysis of the changes in the intrinsic properties of solute molecules with changes in solution condition and is a prerequisite in cation-mediated folding studies of RNA molecules. 

Schuck, P. (2000). Size-distribution analysis of macromolecules by sedimentation velocity ultracentrifugation and Lamm equation modeling. Biophys.J. 78(3), 1606-1619. 

In Eq. (5.5.3) the radial velocity has been replaced by the sedimentation coefficient s, from the definition of Eq. (5.5.2). The fluid dynamicist should be aware that this one-dimensional diffusion equation is known in the ultracentrifuge literature as the Lamm equation (Fujita 1975). In the limit of infinitely dilute solutions D and s are independent of concentration and may be taken out of the derivative to give... 

The evolution of the boundary is described by a partial differential equation, the Lamm equation... 

At equilibrium, by definition, the concentration distribution no longer changes with time, and the Lamm equation is equal to zero. Then, an exponential solution may be found, of the form... 

Molecular weights are determined by solving the equilibrium form of the Lamm equation, generally by nonlinear least-squares fitting of the data to a presumptive model. For a monomeric solute (single ideal species), eqn [6] is a typical model. For associating systems, eqn [6] is extended by terms for each species present in the model. A monomer- -mer system is described by the model... 

Direct boundary fitting Sedimentation velocity data may also be analyzed by least-squares fitting to models based on the Lamm equation. Software packages exist which provide approximate analytical solutions to the equation and which run on desktop computers. Other software packages offer more precise finite element solutions to the equation but require substantial computational power. 

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