### DLS

Dynamic Light Scattering (DLS) of purified protein

DLS, also known as Photon Correlation Spectroscopy, was utilised to measure the protein dimension in solution. This is based on the refraction of laser illuminated particles, at particle angle $$\theta$$, in solution that move according to Brownian motion. It is able to measure as little as 0.1 nm particles. This methodology combines the measured scattering intensity with the electric field equation and allows the Stokes-Einstein equation to be used. The scattering intensity correlation function $$g^2$$ measures the intensity, $$I$$, of a particle at a time, $$t$$, over a period $$\tau$$. The signal changes in increasing duration of measurement and is displayed as an exponential decay. \begin{equation} g^2(q; \tau)=\frac{\big \langle I(t)I(t+\tau)\big \rangle}{\big \langle I(t)\big \rangle ^2}\\ \end{equation} This, in an ideal situation, can be a single decay relating to a monodisperse solution containing 1 type of uniform particle. The equation can be correlated to create the Siegert relationship combining the second order function with a first order autocorrelation function $$g^1$$. \begin{equation} g^2(q; \tau)= 1 + \beta[g^1(q;\tau)]^2\\ \end{equation} The measured $$g^1$$ function can be fitted with an exponential decay. Here, $$\Gamma$$ relates to the diffusion coefficient $$D_t$$ based on the scattering wave vector $$q$$. In this equation, $$\lambda$$ is the laser HeNe wavelength at 632.8 nm, $$n_0$$ is the sample refractive index, $$\theta$$ is the angle of detector location oriented from the sample. \begin{align*} g^1(q; \tau)&= exp\big(-\Gamma\tau\big) \\ \Gamma &= q^2 D_t \\ q &= \frac{4\pi n_0}{\lambda} * sin\Bigg(\frac{\theta}{2}\Bigg) \end{align*} Once the diffusion coefficient is calculated, this can be inserted into the Stokes-Einstein equation. Where $$\eta$$: viscosity in cP, $$\kappa_B$$: Boltzmann’s constant, $$T$$: temperature in Kelvin, $$r$$: hydrodynamic radius. \begin{equation} D = \frac{\kappa_BT}{6\pi\eta r} \end{equation} It is a very effective method to look at protein size in solution and is sensitive enough to detect buffer effects or protein unfolding. This method was applied to verify a monodisperse solution of column-purified TTCF. Zeta potential is obtained from measuring the electrophoretic mobility $$\mu_e$$ in solution when a charge is applied. Then the following equation is utilised: \begin{equation} \mu_e = \frac{V}{E} \end{equation} where $$V$$ = particle velocity (μm/s), $$E$$ = electric field (Volt/cm). This can be processed further with use of the $$Henry$$ equation to determine the $$\zeta$$ potential. \begin{equation} \mu_e = \frac{2\varepsilon_r \varepsilon_0 \zeta f(Ka)}{3\eta} \end{equation} where $$\varepsilon_r$$ = relative permittivity/dielectric constant, $$\varepsilon_0$$ = permittivity of vacuum, $$\zeta$$ = zeta-potential, $$f(Ka)$$ = Henry’s function and $$\eta$$ = viscosity at experimental temperature.

Sample preparation and measurment
Protein was analysed for size and zeta potential. Both measurements required specific sample cells. Polystyrene cuvettes (ZEN0040, Malvern Instruments, UK) were used to measure native solution. Zeta potential measurements were carried out in folded capillary cells (DTS1060, Malvern Instruments, UK) with gold-plated electrodes on each side of the channel. These conduct a small charge through the solution and cause particles to move in the direction of neutral charge. This motion was recorded and correlated with a zeta value. The consensus indicates a stable particle to have a zeta potential > +40 mV or < -40 mV. Particles that fall between this range are susceptible to aggregate with the most unstable particle charge to be around 0 mV.
Protein in buffer at 1 mg / ml was centrifuged at 16,000 x $$g$$ for 20 minutes to remove any particles that could interfere with the measurement, such as dust. The NanoSizer S (Malvern Instruments, UK) was used for measurements. The acquisition protocol is set to measure 10 seconds of counts for 3 repeats. Outputs in size by volume and size by intensity.