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The success and efficiency of sedimentation is highly dependent on molecular weight, concentration, solvent density and the g-force that the sample is subjected to. Here, these variables are considered in the context of sedimentation of various proteins using the ultracentrifuge tools presented in this chapter.

3.6.1 The Concentration and Molecular Weight Requirements for Suc-

cessful Protein Sedimentation

First, the effect of protein concentration and molecular weight on the success of sedi- mentation using the ultracentrifuge tools will be considered. The following equations were published by Bertiniet al. with regards to their ultracentrifuge device and can also be applied to the tools presented here:143

c(h) = cl

Ae−kh2

+ 1 (3.1)

Wherec(h) is the concentration of protein (mol dm-3) ath (the distance from the axis of rotation, in metres) and cl is the limiting concentration of the protein (mol dm-3),

experimentally found to be approximately 700 mg ml-1.149, 150 k and A are given by Equations 3.2 and 3.3.

k= M(1−ρsolvent/ρprotein)ω

2

2RT (3.2)

WhereM is the molecular weight of the protein (kDa),ρsolvent andρprotein are the densi-

ties of the solvent and protein respectively (kg dm-3),ωis the speed of the ultracentrifuge

(rad s-1), R is the universal gas constant andT is the temperature (K).

A is an integration constant relating to the total volume of the device (Vdevice, m3)

and the initial concentration of the solution (c0, mol dm-3):

A=πr₁2 Z b0 b1 c(h)dh+π Z b1 b2 (hp−h+b1 hp r1)2c(h)dh +πr2₂ Z b2 b3 c(h)dh+πr2₃ Z b3 b4 c(h)dh=c0Vdevice (3.3)

Wherer1,r2,r3,b0,b1,b2,b3,b4andhprefer to various measurements of the device (m),

defined in Figure 3.19.

These equations can be used to determine whether the protein can be sedimented straight into the rotor simply by using the MLS-50 ultracentrifuge rotor. This may only be possible for very large proteins, since this rotor can not reach as high speeds as the MLA-150 rotor. The concentration of the protein throughout the device and SSNMR rotor under a range of conditions were calculated using Equations 3.1, 3.2 and 3.3, allowing determination of the fraction of protein in the SSNMR rotor and therefore whether the rotor would be full under these conditions. Figure 3.20 and Table 3.3 show, for a range of molecular weights and proteins, the minimum concentration of protein required for successful sedimentation using both the MLS-50 (black) and MLA-150 (grey) ultracentrifuge tools.

It is clear that high molecular weight proteins (>100 kDa) can be sedimented and packed into a SSNMR rotor using either tool, although the MLA-150 tool may make the process more efficient. On the other hand, it is not possible to sediment

Figure 3.19: A diagram of an ultracentrifuge packing tool highlighting the parameters r1,r2,r3,b0,b1,b2,b3,b4and hp for Equation 3.3. Adapted from Bertiniet al.143

Figure 3.20: Calculated conditions required to fully sediment a protein into a 0.7 mm rotor using the MLS-50 tool at 258,000 x g (black) and the MLA-150 tool at 700,000 x g (grey). The graph indicates the minimum concentration of protein needed for a range of molecular weights.

the lower molecular weight proteins using the MLS-50 tool without unrealistically high concentrations. Therefore the MLA-150 tool is critical for these proteins. For example, sedimentation of a 100 kDa protein requires concentrations of 14.6 and 6.27 mg/ml for the MLS-50 and MLA-150 tools, respectively. Whereas a 25 kDa protein requires 272.70 and 27.51 mg/ml, respectively.

Before deciding which tool to use, these calculations should be performed for the specific protein solution to determine whether the forces produced by the MLS-50 rotor are sufficient to efficiently sediment the sample directly into the rotor or whether the MLA-150 tool is more suitable.

Table 3.3: Examples of proteins with their molecular weights and the minimum concen- trations required for successful sedimentation using the MLS-50 and MLA-150 packing tools at 258,000 x g and 700,000g, respectively.

Protein MW (kDa) Minimum Concentration (mg/ml)

MLS-50 Tool MLA-150 Tool

Ribonuclease A 14.0 480.46 105.45 Chymotrypsinogen A 25.7 263.13 25.53 Ovalbumin 42.9 111.66 8.00 Albumin 69.3 34.88 6.31 Aldolase 157.4 9.86 6.27 Catalase 239.7 9.65 6.27 Apo-ferritin 489.3 9.65 6.27

3.6.2 Calculating the Time Required to Sediment a Protein Sample Using the MLS-50 and MLA-150 Packing Tools

Even in cases where it may be possible to sediment directly into an SSNMR rotor using the MLS-50 ultracentrifuge rotor, it can be significantly more efficient to use the MLA- 150 rotor and a separate packing step. Furthermore, with either tool it is important to predict the time required for sedimentation. To estimate the sedimentation time, the integrated Svedberg equation can be used:143

t= 2.533×1013ln(b4/b0)

Sω2 (3.4)

Where t is the time for complete sedimentation (s), ω is the rotation rate of the ul-

tracentrifuge (rad s-1) and S is the sedimentation coefficient of the protein (10-13 s). The sedimentation coefficient for proteins is typically in the region of 1-20 Svedberg units (1 Svedberg unit = 10-13 s) and can often found in the literature or otherwise determined experimentally. Table 3.4 provides examples of a range of proteins and their sedimentation coefficients.

It is also possible to calculate the maximum sedimentation coefficient (Smax) from

the molecular weight of the protein:

Smax = 0.00361×M2/3 (3.5)

Where M is the molecular weight of the protein in Da. This equation is based on the

assumption that the protein is a smooth sphere, thus the ratio betweenSmax and S can indicate the shape of a protein.

Figure 3.21 indicates the time required to complete sedimentation using the MLS- 50 and MLA-150 tools based on the sedimentation coefficient of the protein. This was calculated using Equation 3.4 and specific examples of sedimentation times can be found

in Table 3.4. It is clear that the greater forces achievable using the MLA-150 tool

substantially reduce the sedimentation time. Note that this calculation does not take into account whether sedimentation is possible or not, thus Equations 3.1, 3.2 and 3.3 are still crucial for determining the minimum protein concentration for successful sedimentation. The exact parameters used for these calculations are reported in Section B.2.