The Partition Coefficient

Chromatography is a core technique of biochemical investigations and is used extensively to purify a protein of interest from a complex mixture such as obtained with the methods described above. Chromatography can be analytical or preparative, but the basis of all forms of chromatography is the distribution or partition coefficient (P), which describes the way in which a compound (the analyte) distributes between two immiscible phases. For two such phases, A and B, the value for this coefficient is a constant at a given temperature and is given by the expression:

The term effective distribution coefficient is defined as the total amount, as distinct from the concentration, of analyte present in one phase divided by the total amount present in the other phase. It is in fact the distribution coefficient multiplied by the ratio of the volumes of the two phases present.

Column Chromatography

Chromatographic systems used for protein purification consist of a stationary phase , which is typically an immobilised solid, and a liquid mobile phase, which is passed through the stationary phase after the mixture of analytes to be separated has been applied to the column. The mobile phase, commonly referred to as the eluent, is passed through the column either by use of a pumping system or gravity (atmospheric pressure). The stationary phase is either coated onto discrete small particles (the matrix) and packed into the column or applied as a thin film to the inside wall of the column. During the chromatographic separation, the analytes continuously pass back and forth between the two phases, exploiting differences in their distribution coefficients, and emerge individually in the eluate as it leaves the column.

Chromatography Components for Protein Purification

A typical chromatographic system suitable for protein purification consists of the following components, ideally be situated in a cold room or refrigerator to maintain protein stability:

 • A stationary phase : Chosen to be appropriate for the analytes to be separated; typically an aqueous buffer with sufficient ioinic strength to maintain a soluble protein

• A column : Research-lab-sized liquid chromatography columns range in length from 5–100 cm, with internal diameters from 4 mm to 6 cm and made of stainless steel, plastic or glass, depending on the application and hence the system pressure involved. The column has to be carefully packed to generate reliable separations. Often, columns are obtained pre-packed from commercial suppliers.

• A mobile phase and delivery system: Chosen to complement the stationary phase and hence to discriminate between the sample analytes and to deliver a constant rate of flow into the column.

• An injector system : To deliver test samples to the top of the column in a reproducible manner.

• A detector with data acquisition : To give a continuous record of the presence of the analytes in the eluate as it emerges from the column. Detection is usually based on the measurement of a physical parameter such as visible or ultraviolet absorption or fluorescence. A typical chromatography system has a computer to control the system and acquire readouts of the monitored physical parameter. The plot of the monitored parameter versus elution time is called a chromatogram .

• A fraction collector : For collecting the separated analytes for further biochemical studies.

Depending on the particle size of the stationary phase (and the desired resolution), the chromatography may be carried out as gravity-driven, low-pressure or high-pressure liquid chromatography. Gravity-driven chromatography is often used with cartridges for a quick single-step purification of small-scale samples and without inline detection systems. Larger samples are processed in pumped systems. In low-pressure liquid chromatography , the flow of the eluent through the column is achieved by a low pressure pump, frequently a peristaltic pump .

The resolution of a mixture of analytes increases as the particle size of the stationary phases decreases, but such a decrease leads to a high back-pressure from the eluent flow. This is addressed by high-pressure liquid chromatography ( HPLC ). This technique is therefore most frequently applied for preparative protein biochemistry and employed in commercially available purification systems that are fully computer-controlled. The much higher resolution of HPLC as compared to low-pressure systems is further surpassed by so-called ultra-performance liquid chromatography ( UPLC) systems, which employ particle sizes of down to 1.7 µm in the stationary phase and back-pressures of up to 150 Mpa.

The application of samples onto HPLC columns in the correct way is a particularly important factor in achieving successful separations. The most common method of sample introduction is by use of a loop injector (Figure 1). This consists of a metal loop, of fixed small volume, that can be filled with the sample. The eluent from the pump is then channelled through the loop by means of a valve switching system and the sample flushed onto the column via the loop outlet without interruption of the f l ow of eluent to the column.

Fig1. HPLC loop injector. (a) The loop is loaded via port 3 with excess sample going to waste via port 5. In this position the eluent from the pump passes to the column via ports 1 and 2. (b) In the injecting position eluent flow is directed through the loop via ports 1 and 6 and then onto the column.

Analyte Development and Elution

 Analyte development and elution relates to the separation of the mixture of analytes applied to the stationary phase by the mobile phase and their elution from the column. Column chromatographic techniques can be subdivided on the basis of the development and elution modes.

• In zonal development , the analytes in the sample are separated on the basis of their distribution coefficients between the stationary and mobile phases. The sample is dissolved in a suitable solvent and applied to the stationary phase as a narrow, discrete band. The mobile phase is then allowed to flow continuously over the stationary phase, resulting in the progressive separation and elution of the sample analytes. If the composition of the mobile phase is constant (as is the case in gas chromatography and some forms of liquid chromatography), the process is called isocratic elution . However, to facilitate separation, the composition of the mobile phase may be gradually changed, for example with respect to pH, salt concentration or polarity. This is referred to as gradient elution. The composition of the mobile phase may be changed continuously or in a stepwise manner.

• In displacement or affinity development (practically only possible in liquid chromatography), the analytes in the sample are separated on the basis of their affinity for the stationary phase. The sample of analytes dissolved in a suitable solvent is applied to the stationary phase as a discrete band. The analytes bind to the stationary phase with a strength determined by their affinity constant for the phase. The analytes are then selectively eluted by using a mobile phase containing a specific solute that has a higher affinity for the stationary phase than have the analytes in the sample. Thus, as the mobile phase is added, this agent displaces the analytes from the stationary phase in a competitive fashion, resulting in their repetitive binding and displacement along the stationary phase and eventual elution from the column in the order of their affinity for the stationary phase, the one with the lowest affinity being eluted first.

Chromatographic Performance Parameters

 The successful chromatographic separation of analytes in a mixture depends upon the selection of the most appropriate process of chromatography, followed by the optimisation of the experimental conditions associated with the separation. Optimisation requires an understanding of the processes that occur during the development and elution, and of the calculation of a number of experimental parameters characterising the behaviour of each analyte in the mixture.

In any chromatographic separation, two processes occur concurrently and affect the behaviour of each analyte and hence the success of the separation of the analytes from each other. The first involves the basic mechanisms defining the chromatographic process such as adsorption, partition, ion exchange, ion pairing and molecular exclusion. These mechanisms involve the unique kinetic and thermodynamic processes that characterise the interaction of each analyte with the stationary phase. The second general process comprises diffusion and non-specific interactions, which tend to oppose the separation and result in non-ideal behaviour of each analyte. These processes manifest themselves as a broadening and tailing of each analyte peak. The challenge is to minimise these secondary processes.

Retention Time

A chromatogram is a pictorial record of the detector response as a function of elution volume or retention time. It consists of a series of peaks, ideally symmetrical in shape, representing the elution of individual analytes, as shown in Figure 2. The retention time t R for each analyte has two components. The first is the time it takes the analyte molecules to pass through the free spaces between the particles of the matrix coated with the stationary phase. This time is referred to as the dead time, t M. The volume of the free space is referred to as the column void volume , V 0 . The value of t M will be the same for all analytes and can be measured by using an analyte that does not interact with the stationary phase, but simply spends all of the elution time in the mobile phase travelling through the void volume. The second component is the time the stationary phase retains the analyte, referred to as the adjusted retention time , t´ R . This time is characteristic of the analyte and is the difference between the observed retention time and the dead time:

Fig2. (a) Chromatogram of two analytes showing complete resolution and the calculation of retention times; (b) chromatogram of two analytes showing incomplete resolution (fused peaks); (c) chromatogram of an analyte showing excessive tailing, For explanation of parameters shown see main text.

Retention Factor

One of the most important parameters in column chromatography is the retention factor, k (previously called capacity factor). It is simply the additional time that the analyte takes to elute from the column relative to an unretained or excluded analyte that does not interact with the stationary phase and which, by definition, has a k value of 0. Thus:

Note that k has no units.

It is apparent from this equation that if the analyte spends an equal time in the stationary and mobile phases, its t R would equal 2 ×  t M and its k would thus be 1.

Similarly, if it spent four times as long in the stationary phase as the mobile phase, t R would equal 5 ×  t M and so the retention factor would be:

If an analyte has a k of 4, it follows that there will be four times the amount of analyte in the stationary phase than in the mobile phase at any point in the column at any time. It is evident, therefore, that k is related to the distribution coefficient of the analyte, which was defined as the relative concentrations of the analyte between the two phases. Since amount and concentration are related by volume, we can write:

where m S is the mass of analyte in the stationary phase, m M is the mass of analyte in the mobile phase, V S is the volume of stationary phase and V M is the volume of mobile phase. The ratio V S / V M is referred to as the volumetric phase ratio , β. Hence:

A larger retention factor results in the distribution ratio favouring the stationary phase, which increases retention time. Therefore, the retention factor for an ana lyte will increase with both the distribution coefficient between the two phases and the volume of the stationary phase. Values of k normally range from 1 to 10. Retention factors are important, because they are independent of the physical dimensions of the column and the rate of fl ow of mobile phase through it. They can therefore be used to compare the behaviour of an analyte in different chromatographic systems. They are also a reflection of the selectivity of the system that, in turn, is a measure of its inherent ability to discriminate between two analytes. Such selectivity is expressed by the selectivity or separation factor , α, which can also be viewed as simply the relative retention ratio for the two analytes:

The selectivity factor is influenced by the chemical nature of the stationary and mobile phases. Some chromatographic mechanisms are inherently highly selective, for example affinity chromatography.

Plate Height

Chromatography columns are considered to consist of a number of adjacent zones in each of which there is sufficient space for an analyte to completely equilibrate between the two phases. Each zone is called a theoretical plate (of which there are N in total in the column). The length of column containing one theoretical plate is referred to as the plate height, H , which has units of length (normally micrometres).

The smaller the value of H and the greater the value of N , the more efficient the column in separating a mixture of analytes. The numerical value of both N and H for a particular column is expressed by reference to a particular analyte. Plate height is simply related to the width of the analyte peak, expressed in terms of its standard deviation σ ( Figure 2), and the distance it travelled within the column, x . Specifically:

For symmetrical Gaussian peaks, the base width is equal to 4 × σ and the peak width at the point of inflection is equal to 2 × σ. Hence, the value of H can be calculated from the chromatogram by measuring the peak width. The number of theoretical plates in the whole column of length L is equal to L divided by the plate height:

If the position of a peak emerging from the column is such that x = L , Equation 5.9 can be converted to

knowing that the width of the peak at its base, w , is equal to 4 × σ and hence σ = ¼ × w .

If both L and w are measured in units of time rather than length, then Equation 5.10 becomes:

Rather than expressing N in terms of the peak base width, it is possible to express it in terms of the peak width at half height ( full width at half maximum , FWHM ) and this has the practical advantage that this is more easily measured:

Equations 5.11 and 5.12 represent alternative ways to calculate the column efficiency in theoretical plates. The value of N , which has no units, can be as high as 50 000 to 100 000 permetre for efficient columns and the corresponding value of H can be as little as a few micrometres. The smaller the plate height (the larger the value of N ), the narrower the analyte peak ( Figure 3 ).

Fig3. Relationship between the number of theoretical plates ( N ) and the shape of the analyte peak.

Peak Broadening

A number of processes oppose the formation of a narrow analyte peak, thereby increasing the plate height:

• Application of the sample to the column : It takes a finite time to apply the analyte mixture to the column, so that the part of the sample applied first will already be moving along the column by the time the final part is applied. The part of the sample applied first will elute at the front of the peak.

• Longitudinal diffusion : Fick’s law of diffusion states that an analyte will diffuse from a region of high concentration to one of low concentration at a rate determined by the concentration gradient between the two regions and the diffusion coefficient of the analyte. Thus, the analyte within a narrow band will tend to diffuse outwards from the centre of the band, resulting in band broadening.

• Multiple pathways : The random packing of the particles in the column results in the availability of many routes between the particles for both mobile phase and analytes. These pathways will vary slightly in length and hence elution time; the smaller the particle size the less serious this problem.

• Equilibration time between the two phases : It takes a finite time for each analyte in the sample to equilibrate between the stationary and mobile phases as it passes down the column. As a direct consequence of the distribution coefficient ( P ), some of each analyte is retained by the stationary phase, whilst the remainder stays in the mobile phase and continues its passage down the column. This partitioning automatically results in some spreading of each analyte band. Equilibration time, and hence band broadening, is also influenced by the particle size of the stationary phase. The smaller the size, the less time it takes to establish equilibration.

Two of these four factors promoting the broadening of the analyte band are influenced by the flow rate of the eluent through the column. Longitudinal diffusion, defined by Fick’s law, is inversely proportional to flow rate, whilst equilibration time due to the partitioning of the analyte is directly proportional to fl ow rate. These two factors together with the problem of multiple pathways affect the value of the plate height ( H ) for a particular column and, as previously stated, plate height determines the width of the analyte peak. The precise relationship between the three factors and plate height is expressed by the van Deemter equation ( Equation 5.13 ), which is illustrated graphically in Figure 4:

where u is the flow rate of the eluent (in units of m s −1 ) and A, B and C are constants for a particular column and stationary phase relating to multiple paths ( A ), longitudinal diffusion ( B ) and equilibration time ( C ).

Fig4. Illustration of the van Deemter equation. The plot shows that the optimum flow rate for a given column is the net result of the influence of flow rate on longitudinal diffusion, equilibration time and multiple pathways.

Figure 4 gives a clear demonstration of the importance of establishing the optimum flow rate for a particular column. Longitudinal diffusion is much faster in a gas than in a liquid and, as a consequence, flow rates are higher in gas chromatography than in liquid chromatography.

As previously stated, the width of an analyte peak is expressed in terms of the standard deviation σ, which is half the peak width at the point of inflection (0.607 ×  h p , where h p is the peak height; see Figure 2 ). It can be shown that

where D is the diffusion coefficient of the analyte. The diffusion coefficient is a measure of the rate at which the analyte moves randomly in the mobile phase from a region of high concentration to one of lower concentration; it has units of m²  s −1.

Since the value of σ is proportional to the square root of t R , it follows that if the elution time increases by a factor of four, the width of the peak will double. Thus, the longer it takes a given analyte to elute, the wider will be its peak. This phenomenon counteracts the improvement in resolution achieved by increasing the column length ( Equation 5.16 ).

Asymmetric Peaks

In some chromatographic separations, the ideal Gaussian-shaped peaks are not obtained, but rather asymmetrical peaks are produced. In cases where there is a gradual rise at the front of the peak and a sharp fall after the peak, the phenomenon is known as fronting . The most common cause of fronting is overloading the column, so that reducing the amount of mixture applied to the column often resolves the problem. In cases where the rise in the peak is normal, but the tail is protracted, the phenomenon is known as tailing (see Figure 2 c). The probable explanation for tailing is the retention of analyte by a few sites (frequently hydroxyl groups) on the stationary phase, commonly on the inert support matrix. Such sites strongly adsorb molecules of the analyte and only slowly release them. This problem can be overcome by chemically removing the sites, for example by treating the matrix with a silanising reagent such as hexamethyldisilazine. This process is sometimes referred to as cap ping . Peak asymmetry is usually expressed as the ratio between the width of the peak at the centre (i.e. at full peak height h p ) and the width of the peak at 0.1 × h p .

Resolution

The success of a chromatographic separation is judged by the ability of the system to resolve one analyte peak from another. Resolution ( R S ) is defined as the ratio of the difference in retention time (∆ t R ) between the two peaks to the mean of their base widths ( w A and w B ), w av

When R S = 1.0, the separation of the two peaks is 97.7% complete (thus the overlap is 2.3%). When R S = 1.5, the overlap is reduced to 0.2%. Unresolved peaks are referred to as fused peaks ( Figure 2b ). Provided the overlap is not excessive, the analysis of the individual peaks can be made on the assumption that their characteristics are not affected by the incomplete resolution.

Resolution is influenced by column efficiency, selectivity factors and retention factors according to:

where k 2 is the retention factor for the longest retained peak and k av is the mean retention factor for the two analytes. Equation 5.16 is one of the most important in chromatography as it enables a rational approach to the improvement of the resolution between analytes. For example, it can be seen that resolution increases with the square root of N . Since N is linked to the length of the column, doubling the length of the column will increase resolution by √2 ≈ 1.4 and in general is not the preferred way to improve resolution. Since both retention factors and selectivity factors are linked to retention times and retention volumes, altering the nature of the two phases or their relative volumes will impact on resolution. Retention factors are also dependent upon distribution coefficients, which in turn are temperature dependent; hence altering the column temperature may improve resolution.

The capacity of a particular chromatographic separation is a measure of the amount of material that can be resolved into its components without causing peak overlap or fronting. Ion-exchange chromatography resins have a high capacity, which is why it this type of chromatography is often used in the earlier stages of a purification process.

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