Which of the following phases of the bacterial growth curve is matched with the correct definition?

11.2 Batch Growth Patterns

When a liquid nutrient medium is inoculated with a seed culture (inoculums), the organisms selectively take up dissolved nutrients from the medium and convert them into biomass. A typical batch growth curve includes the following phases: (1) lag phase, (2) logarithmic or exponential growth phase, (3) deceleration phase, (4) stationary phase, and (5) death phase. Fig. 11.1 describes a batch mammalian cell growth cycle (which is also typical for a microbial cell). While several parameters are shown, the cell count (VCD) or cell biomass is in focus for growth regime classification.

The lag phase occurs immediately after inoculation and is a period of adaptation of cells to a new environment. Microorganisms reorganize their molecular constituents when they are transferred to a new medium. Depending on the composition of nutrients, new enzymes are synthesized, the synthesis of some other enzymes is repressed, and the internal machinery of cells is adapted to the new environmental conditions. These changes reflect the intracellular mechanisms for the regulation of the metabolic processes discussed in Chapter 9. During this phase, cell mass may increase a little, without an increase in cell number density. When the inoculum is small and has a low fraction of cells that are viable, there may be a pseudolag phase, which is a result not of an adaptation but of a small inoculum’s size or poor condition.

Low concentration of some nutrients and growth factors may also cause a long lag phase. For example, the lag phase of Enterobacter aerogenes (formerly Aerobacter aerogenes) grown in a glucose and phosphate buffer medium increases as the concentration of Mg2 +, which is an activator of the enzyme phosphatase, is decreased. As another example, even heterotrophic cells require CO2 fixation (to supplement intermediates removed from key energy-producing metabolic cycles during rapid biosynthesis), and excessive sparging can remove metabolically generated CO2 too rapidly for cellular restructuring to be accomplished efficiently, particularly with a small inoculum.

The age of the inoculum culture has a strong effect on the length of lag phase. The age refers to how long a culture has been maintained in a batch culture. Usually, the lag period increases with the age of the inoculum. In some cases, there is an optimal inoculums age resulting in minimum lag period. To minimize the duration of the lag phase, cells should be adapted to the growth medium and conditions before inoculation, and cells should be young (or exponential phase cells) and active, and the inoculum size should be large (5–10% by volume). The nutrient medium may need to be optimized and certain growth factors included to minimize the lag phase. Many commercial fermentation plants rely on batch culture; to obtain high productivity from a fixed plant size, the lag phase must be as short as possible.

Multiple lag phases may be observed when the medium contains more than one carbon source. This phenomenon, known as diauxic growth, is caused by a shift in metabolic pathways in the middle of a growth cycle (see Example 10.1). After one carbon source is exhausted, the cells adapt their metabolic activities to utilize the second carbon source. The first carbon source is more readily utilizable than the second, and the presence of more readily available carbon source represses the synthesis of the enzymes required for the metabolism of the second substrate.

The maximum growth phase is also known as the exponential growth phase or logarithmic growth phase. In this phase, the cells have adjusted to their new environment. After this adaptation period, cells can multiply rapidly at a maximum rate, and cell mass and cell number density increase exponentially with time. This is a period of balanced growth, in which all components of a cell grow at the same rate (pseudosteady state). That is, the average composition of a single cell remains approximately constant during this phase of growth. During balanced growth, the net specific growth rate determined from either cell number or cell mass would be the same. The specific growth rate is constant, from which a phenomenological model is proposed for the exponential growth phase:

(11.2)rX=μnetX

with the net specific growth rate being constant during this growth phase. This simple relation of Eq. (11.2) is called the Malthus growth model. In the batch process, the rate of change of biomass concentration is the same as the rate of generation of biomass (mass balance). Integration of the mass balance equation with Eq. (11.2) as the rate of generation of biomass yields:

(11.3a)lnXX 0=μnett

or

(11.3b)X=X0eμnett

where X and X0 are cell concentrations at time t and initial time t = 0; respectively.

The time required to double the microbial mass can be computed using Eq. (11.3a) as:

(11.4)td=ln2μnet

The doubling time is also the time required for a new generation of cells to appear during exponential growth period.

The deceleration growth phase follows the maximum growth phase. In this phase, growth decelerates due to either depletion of one or more essential nutrients or the accumulation of toxic byproducts of growth. For a typical bacterial culture, these changes occur over a very short period of time. The rapidly changing environment results in unbalanced growth. During unbalanced growth, cell composition and size will change. In the exponential phase, the cellular metabolic control system is set to achieve maximum rates of reproduction. In the deceleration phase, the stresses induced by nutrient depletion or waste accumulation cause a restructuring of the cell to increase the prospects of cellular survival in a hostile environment. These observable changes are the result of the molecular mechanisms of repression and induction that we discussed in Chapter 9. Because of the rapidity of these changes, cell physiology under conditions of nutrient limitation is more easily studied in continuous culture, as discussed later in Chapter 12.

Malthus growth model is valid only in the exponential growth phase. A modification of the Malthus model by Verhulst in 1844 included an apparent biomass inhibition term:

(11.5) rX=kX1−XX∞

where X∞ is the carrying capacity of the cells in the medium and k is the carrying capacity coefficient. For a batch growth of constant culture volume, cell balance subjected to growth rate given by Eq. (11.5) gives rise to

(11.6)X=X0ekt1−X0X∞1−ekt

Eq. (11.6), which is also termed the Logistic Equation. The Verhulst model is able to describe the exponential growth phase, the deceleration phase, and the stationary phase, via Eq. (11.6). Therefore, the Verhulst model (or logistic model) is a more accurate phenomenological model than the Malthus model.

The stationary phase starts at the end of the deceleration phase, when the net growth rate is zero (no cell division) or when the growth rate is equal to the death rate. Even though the net growth rate is zero during the stationary phase, cells are still metabolically active and produce secondary metabolites. Primary metabolites are growth-related products and secondary metabolites are not growth related. In fact, the production of certain metabolites is enhanced during the stationary phase (eg, antibiotics, some hormones) due to metabolite deregulation. During the course of the stationary phase, one or more of the following phenomena may take place:

Total cell mass concentration may stay constant, but the number of viable cells may decrease.

Cell lysis may occur and viable cell mass may drop. A second growth phase may occur and cells may grow on lysis products of lysed cells (cryptic growth).

Cells may not be growing but may have active metabolism to produce secondary metabolites. Cellular regulation changes when concentrations of certain metabolites (eg, carbon, nitrogen, and phosphate) are low. Secondary metabolites are produced as a result of metabolite deregulation.

During the stationary phase, the cell catabolizes cellular reserves for new building blocks and for energy-producing monomers. This is called endogenous metabolism. The cell must always expend energy to maintain an energized membrane (ie, a proton-motive force) and transport of nutrients for essential metabolic functions such as motility and repair of damage to cellular structures. This energy expenditure is called maintenance energy. As such, maintenance energy and endogenous metabolism are not limited to the stationary phase, but become dominant in the stationary phase. The maintenance or endogenous expenditure is just a small fraction of the total cell needs during maximum growth. When the primary metabolism diminishes as in the stationary phase, the endogenous metabolism becomes dominant.

The reason for termination of growth may be either exhaustion of an essential nutrient or accumulation of toxic products. If an inhibitory product is produced and accumulates in the medium, the growth rate will slow down, depending on inhibitor production, and at a certain level of inhibitor concentration, growth will stop. Ethanol production by yeast is an example of a fermentation in which the product is inhibitory to growth. Dilution of a toxified medium, addition of an unmetabolizable chemical compound complex with the toxin, or simultaneous removal of the toxin would alleviate the adverse effects of the toxin and yield further growth.

The death phase (or decline phase) follows the stationary phase. However, some cell death may start during or even before the stationary phase, and a clear demarcation between these two phases is not always possible. Often, dead cells lyse, and intracellular nutrients released into the medium are used by the living organisms during stationary phase. At the end of the stationary phase, because of either nutrient depletion or toxic product accumulation, the death phase begins. The death rate can be thought of as a first-order reaction. Because S is zero, μG is zero starting from the stationary phase:

(11.7) rX=−kdX

where kd is a first-order rate constant for cell death. Mass balance of the cell biomass in the batch reactor leads to:

(11.8)−kdXV=rXV=dXVdt

which can be integrated to yield (for constant medium V):

(11.9)X=XS0e−kdt

where XS0 is the cell mass concentration at the beginning of the stationary phase.

During the death phase, cells may or may not lyse, and the reestablishment of the culture may be possible in the early death phase if cells are transferred into a nutrient-rich medium. In both the death and stationary phases, it is important to recognize that there is a distribution of properties among individuals in a population. With a narrow distribution, cell death will occur nearly simultaneously; with a broad distribution, a subfraction of the population may survive for an extended period. It is this subfraction that would dominate the reestablishment of a culture from inoculum derived from stationary or death-phase cultures. Thus, using an old inoculum may select for variants of the original strain having altered metabolic capabilities.

While the phenomenological models can describe the batch experiments of cell growth reasonably well, the parameters are not as meaningful for further genetical and more mechanistic evaluations. The regimes, especially the exponential phase, deceleration phase, and stationary phase, change with different loading of the same nutrients. The cell growth is understandably related to the availability of the substrates (or nutrients) in the medium, which may not be due to the biomass inhibition as the logistic model depicts. One can imagine that the exponential growth is due to the sufficient supply of nutrients. The stationary phase is due to the exhaustion of the nutrients, not necessarily due to cell biomass inhibition. Therefore, by not examining the substrate change, it leads to an incomplete description of cell growth.

2.52.3.3 Chiral Stationary Phases

The CSP constitutes the basis of chiral chromatography. Many different materials are available for all kinds of different separations. Here only a short explanation can be given on chiral recognition mechanisms, followed by a brief survey of the most important CSPs.

Concerning the induction of chirality, chiral chromatography is a direct method. The exploited chiral interaction is the formation of transient diastereomeric complexes between the analyte and the CSP. This can be illustrated using Pirkle’s three-point interaction principle illustrated in Figure 8. According to the principle, at least three interactions are required, of which at least one needs to be stereochemically dependent. The interactions involved can be electrostatic interactions, ion exchange, π–π-interactions, hydrogen bonds, or van der Waals forces. The formed associates are not very stable; flushing with mobile phase suffices for cleavage. Therefore, chiral chromatography is usually operated in plain elution mode. Solvent gradients are rarely required. Adsorption/desorption cycles do not play a role.

Note that chiral and achiral interactions occur simultaneously. The typical pronounced peak tailing of the stronger adsorbing enantiomer (Figure 7(b)) can be attributed to chiral adsorption sites with low capacity and high energy (additionally, steric effects can play a role for this tailing). Retention itself is often influenced significantly by a nonselective interaction between the solute and the achiral support.

CSPs can be classified in different ways. Table 1 contains the most common CSP classification based on the type of chiral interaction.

Table 1. Chiral stationary phases and the interactions involved

Polymeric phases based on polysaccharides are commercially the most relevant. Examples are modified cellulose and amylose, or tartaric acid derivatives. Such phases are available as pure material or coated on an achiral support to increase stability. These CSPs are versatile and offer, due to the large amount of chiral material, high loading capacities. Another important class are so-called Pirkle or brush-type CSPs that contain small selector molecules, often amino acid derivatives, covalently bound to a support (typically silica gel). Also cyclodextrins, cyclic oligosaccharides with a strongly hydrophobic cavity confined by a more hydrophilic opening, are commonly used. Their structure allows for the formation of inclusion complexes with various molecules. Usually operated under reversed-phase conditions, enantioselectivity can be adjusted by varying pH and the amount and type of organic modifiers. The final type to be mentioned are CSPs based on macrocyclic glycopeptides. Their antibiotic selectors offer several chiral cavities offering multiple possibilities for interaction.

In cases where no phase is available for a given separation, it can be an option to synthesize a custom CSP. An interesting possibility in this context is to use one of the enantiomers as a template for a molecularly imprinted CSP (see also Section 2.52.6 on molecularly imprinted membranes).

2.1 New Liquid Chromatography Separation Modes

Apart from the advances in mass spectrometry (MS) techniques that have had a significant impact on the field of pharmaceuticals in environmental studies, new columns for chromatography also have made a significant contribution. Pharmaceutical compounds, being more polar and less volatile, are typically better retained on reversed-phase (RP) columns. However, some modifications in the stationary phase of the column made possible for different range of compounds to be better retained than in C18 columns. Other advances in chromatography columns are monolithic columns, which can be a very good alternative to particle-packed columns in terms of separation efficiency [12–14]. Since they typically have small-sized skeletons and wide through pores, much higher separation efficiency can be achieved than in the case of particle-packed columns at a similar pressure drop [15]. One advantage of these columns is that they can work at high flow rates (up to 10 mL/min) in conventional column lengths (4.6 mm ID) without generating high back pressures. [13]. On one hand, technological advances in sorbent materials gave better performance in terms of efficiency and fast liquid chromatography (LC) using RP columns packed with sub-2 μm particles [16]. These ultrahigh-pressure LC columns contribute to faster analysis in many standard applications. However, for research on TPs of pharmaceuticals, there is a chance that the TPs are too polar to be retained on RP columns. This issue can be solved using hydrophilic interaction chromatography (HILIC) columns. HILIC mode of separation relies on polar stationary phases (such as silica gel or aminopropyl HILIC columns) and aqueous–organic mobile phases rich in organic solvents (usually methanol, acetonitrile, or their mixtures) in which water is introduced to play the role of a stronger eluting solvent [13,17]. When more than 1% of water is used in the mobile phases, the layer of water adsorbed on the polar stationary phase is usually thick enough to allow for liquid–liquid partitioning between the bulk mobile phase and the adsorbed aqueous layer. HILIC retention is controlled by a combination of partition and other interactions such as ion exchange, H bonding, and dipole–dipole affecting the selectivity of the separation [18,19]. One advantage of this is that in HILIC mode, analytes elute in a reverse order as compared to RP chromatography and ion-pairing agents needed for retention in RP columns are not necessary, thereby making it easier to couple to MS. On the other hand, high percentage of organic solvent (acetonitrile) enhances ionization and thus increases sensitivity [13]. HILIC proved to be efficient in the determination of the photolysis products of the antiviral zanamivir which was better retained than in RP-LC and it was possible to separate zanamivir from an isobaric TP [20].

Reverse phase chromatography (RPC)

When the stationary phase has greater polarity than the mobile phase it is termed “normal phase chromatography.” When the opposite is the case, it is termed “reverse phase chromatography.” RPC utilizes a solid phase (eg, silica) which is modified so as to replace hydrophilic groups with hydrophobic alkyl chains. This allows the separation of proteins according to their hydrophobicity. More-hydrophobic proteins bind most strongly to the stationary phase and are therefore eluted later than less-hydrophobic proteins. The alkyl groupings are normally eight or eighteen carbons in length (C8 and C18). RPC can also be combined with affinity techniques in the separation of, for example, proteins and peptides (Davankov, Kurganov, & Unger, 1990).

Curdlan

In stationary phase growth, A. faecalis secretes an exopolysaccharide composed of linear, unbranched d-glucose molecules in a β-1,3 glycosidic linkage. This form of polysaccharide is synthesized by several bacterial species and is known by the common name curdlan. Synthesis of curdlan is believed to occur through the polymerization of UDP-glucose units, and two loci involved in curdlan synthesis have been cloned from A. faecalis. Curdlan has potential as a food additive and may even have medical applications.

The property of curdlan that has the most promise in regard to the food industry is the ability of the polysaccharide to form a stable gel. In aqueous solution, curdlan is insoluble, but it becomes soluble upon heating. Increasing the pH or additional heating of a curdlan solution causes a change of phase to a solid gel. This change is the result of the previously disordered glucan chains, assuming an ordered triple helical structure. The gel exhibits stability across a wide pH range, and retains its physical properties on freezing and thawing. Curdlan currently is used in Japan as a food stabilizer and thickener.

Curdlan also exhibits properties that may prove useful in a clinical setting. A sulfated form of the polysaccharide has been shown to prevent human immunodeficiency virus from binding to the CD4 receptor of T cells in vitro. Clinical trials are under way to demonstrate in vivo effectiveness. Certain viral and bacterial infections are known to cause a rise in levels of the hormone tumor necrosis factor (TNF). TNF normally is involved in the host immune response, but in some disease states, a rise in the hormone is observed, and this causes problems such as inflammation and endotoxic shock. Curdlan sulfate has been found to prevent exaggerated levels of TNF expression, retaining the beneficial action of the hormone without the side effects.

5.6.2 Sound From the Locally-Driven Fluid-Loaded Plate

The stationary phase solution for the far-field sound, at k0r>>1, can be evaluated using Eq. (5.40e) by substituting the wave numbers of stationary phase, Eq. (5.77), into Eqs. (5.54a) and (5.96). Using Eqs. (5.36b), (5.45) and (5.40e) we obtain the modal excitation pressure

Pm n(ω)=[FAp] ψmn(x→0)

for the point force spectrum. The resulting modal radiated sound pressure re pref=1 μPa at 1 m. We use the summation over all m,n modes to give the total pressure.

(5.102)Pmn(r,θ,ϕ,ω)rprefF(ω)=ρ0ω2π1 ApΨm(x1o)Ψn(x3 o)Smn(k1¯,k3¯)Zmn(ω,k1¯,k3¯) and|Pa(r,θ,ϕ,ω)r prefF(ω)|≈|∑mnρ0 ω2π1ApΨm(x1o )Ψn(x3o)Smn(k1¯,k3¯)Zmn(ω,k1¯,k3¯)|

Fig. 5.23 shows examples of both the modal sound pressure and the total of the modal sum sound pressure scaled as above for individual modes as a function of normalized frequency, ωL1/2πc0=k0L1/2π. The drive is 1 N applied to the plate, water is on one side of the plate. The values of sound pressure are all for the indicated values of lateral mode orders and shown as continuous functions of longitudinal mode wave numbers and frequency expressed as the acoustic wave number. Both of these are normalized on the length of the panel, L1. Radiated sound pressure is evaluated at θ=45° and ϕ=45° off the normal to the plate, all dB re 1 μPa. The top three illustrations are modal sound pressures for m, n=(m, 1), (m, 3), and (m, 7), respectively. The plate is driven off center in order to simultaneously excite as many modes as possible. The bottom illustration plots all modal sound pressures for (m,7) modes as well as total pressure summed over all modes (solid heavy line); these are compared with the sound from a point dipole applied to an infinite plate (the dashed line, which will be discussed in some detail below).

The spectra shown in Fig. 5.23 illustrates a number of important structural acoustic mechanisms. In each of the top 3 two dimensional spectra shown in color scale are identified the straight line traces of the acoustic coincidence, km=k0 near the k0 coordinate. Highlights are apparent as this coincidence passes through the various resonances. The wave number pass bands of the various mode shapes (discussed previously in connection with Fig. 5.9) are shown by the scalloped patterns in the figures. Of these modes, the resonant ones are highlighted by enhanced radiated sound. The upper boundary of the (k0, km) patterns for each lateral wave number, kn, is determined by the quadratic function relating frequency to plate wave number, the |k|=kp line as noted. Finally, as shown in the lower line graph, the knL3/π=7 modes are apparently responsible for the sound at approximately k0/2π=0.6 and approximately 26 where, for this value of kn, km=kp at a plate resonance and km=k0 at acoustic coincidence for this mode.

In the case of the point-driven infinite plate without boundaries, and of course no modes, the expression that is equivalent to Eq. (5.102) effectively replaces Ψ m(x1o)Ψn( x3o) by unity and Smn(k1¯,k3¯) by Ap in the limit of k0<kp; i.e., well below acoustic coincidence for the plate bending,

(5.103a)Pa(r,θ,ϕ,ω)rp0F(ω)≈∑mn ρ0ω2π1Zmn(ω,k1¯,k3¯)eik0r

or

(5.103b)Pa(r,θ,ϕ,ω)rp0F(ω)≈−iρ0ω2π1[msω−iρ0ωk0cosϕ ]eik0r

This equation is identical to the classically known function [66–68]

(5.104)p(r,ϕ,t)=−ik0F02πβcosϕcosϕ−iβ ei(k0r−ωt)r

and

(5.105)β=ρ0c0/msω

is the fluid loading factor for fluid on one side.

If the fluid was on both sides, then 2β replaces β in Eq. (5.104). Eq. (5.103) applies when the area over which the force extends is smaller than the bending wavelength, and when the frequency of excitation is below coincidence so that kp>k0. Eq. (5.103b) is the dashed line in Fig. 5.23 and forms somewhat of a mean line with the contributions from specific modes causing excursions above and below the values for the infinite plate. When a large (infinite) plate is driven at a point at low frequencies, the sound radiated is independent of the plate material and thickness (β is large) and it is double that emitted by just the force as a free dipole in the water. This is shown in Fig. 5.24 where several example cases are plotted for various plate thicknesses to illustrate the point. To this point, compare this expression to Eq. (2.75) for the dipole radiation from a point force in an unbounded fluid. When β becomes small (less than unity) the force on the plate ceases to act as an amplified free dipole and takes on a quite different character.

When we add complexity in the form of boundaries (as with a simple-supported plate in a rigid baffle) and thereby introduce modes, the picture changes by the introduction of resonance character to the spectrum of the sound and vibration as discussed in connection with Figs. 5.21 and 5.22. Fig. 5.25 further illustrates this in presenting the modal sum of all contributing modes for the same example of a simple supported plate that was discussed above. This is an example of a plate that has a high mode density in the frequency range that we are discussing. Upon summing all modes, three distinct frequency regions emerge for which the physics of sound generation changes. At low frequencies, below the fundamental mode of the plate, the sound increases as (frequency)4 since the displacement pattern of the plate is dominated by a simple half-wave for which the plate displacement represents a net volume change on the fluid side. At frequencies between fundamental resonance and that for which k0L1/π~1 the plate modes dominate the sound character. For frequencies for which k0L1/π>1 the plate behaves as infinite, see also Fig. 5.22 and the fluid loading factor also begins its approach toward unity. In this upper frequency range, the radiated sound also behaves as if the plate is infinite. At these high frequencies the plate is beginning to appear acoustically large as well as noted by the convergence of the two lines.

13.10.2 Enantioseparations in capillaries packed with achiral stationary phases in combination with chiral buffer additives

Enantioseparations with achiral stationary phases in combination with CMPAs was rather popular in HPLC in early days when highly effective CSPs were not yet available. This mode in CEC may offer some alternative possibilities compared to a mode with CSPs [8]. For example, a chiral additive and its concentration in the mobile phase may be varied easier compared to a CSP packed into the capillary. One capillary with a standard packing material may be used in combination with many different additives to the BGE. Thus, it is somewhat possible to find some niches for chiral CEC with achiral stationary phase in combination with chiral additives of a BGE compared to chiral CEC with CSPs. It is rather difficult to do this when comparing this technique to chiral CE because the chiral selectors which are soluble and exhibit chiral recognition ability in a given BGE is easier and in general more effective to be used in the capillary filled with electrolyte compared to the capillary packed with achiral packing material. However, even in this case, some particular applications may be found for the technique described in this section because the chiral selector preadsorbed on the achiral stationary phase certainly behaves differently compared to that residing in free solution.

Two interesting works on the effect of rather low applied voltage on the behavior of a CMPA in the HPLC mode have been published [125,126]. Porter and coworkers [125,126] created a dynamically controlled separation system consisting of a porous graphitic carbon stationary phase and β-CD. These authors have found that the applied voltage affects the amount of β-CD electrosorbed on the surface of a conductive support. This was undoubtedly confirmed by switching the enantiomer elution order of mephenytoin depending on the applied voltage. At the lower voltage, the amount of the chiral selector residing in the mobile phase was higher and this determined the enantiomer elution order. In contrast, at higher voltage, the amount of electrosorbed β-CD on the stationary surface was higher and the enantiomer elution order was determined by this fraction. The voltage used in these studies was low and basically not used as a driving force for the analytes. Therefore, the authors named this technique as electrochemically modulated liquid chromatography. A combination of this technique with CEC driving mechanism appears to be promising also from the viewpoint of mechanistic studies.

16.3.2 Classification of Chromatography Stationary Phases

Another aspect of a chromatography stationary phase relates to its physical attributes. A classification based on these properties has been proposed [15], and is shown here in a slightly modified form in Fig. 16.3. The classification is based on a division of materials into discrete and continuous stationary phases. The former includes particles of different shapes and sizes, and the latter monoliths and activated membranes. The continuous phases are either formed in situ in the case of monoliths [16], or in the case of membranes, they are cast as long sheets or extruded as hollow fibers, which are packed later into relevant cartridges of predefined sizes. Recently introduced nano-fibers [17] can be also classified in this category, as their manufacturing process resembles preparation of a membrane cartridge. With respect to pore size and pore size distribution, continuous stationary phases typically belong to the family of either macroporous (rp > 50 nm), or microporous/mesoporous (2 nm < rp < 50 nm) materials. In either case, the macropores must form a percolating network of pores as they act as channels through which fluid is transported via convection. In practice, these pores are frequently in the micro-meter range (e.g., rp ≈ 1–10 μm) [18]. Compared with typical discrete porous stationary phases, macroporous monoliths/membranes have much lower surface areas and thus much lower binding capacities for small solutes. The presence of mesopores partially eliminates this disadvantage.

The discrete stationary phases can be used either in packed (column) or dispersed (batch) form. They are made of particles of specific form/shape and defined (characteristic) size. In the case of packed column format, the interstitial pores (i.e., the voids between packed particles), are responsible for providing a percolating path for transport of liquid through the packed column (Fig. 16.4). As already mentioned, the amount or resin packed in a column of a given volume is most conveniently expressed through the column interstitial, or bed, porosity, εint or εb, (i.e., the fraction of column volume occupied by the interstitial pores). Size of the interstitial pores can be estimated from characteristic dimension of the particles packed and the porosity of the packed bed (εb). For instance, for spherical particles of diameter dp and rhombohedral packing (εb = 0.26), the smallest opening between particles has a diameter equal to one-third of the particle diameter. It is also important to mention at this point that particle size has no effect on the value of interstitial porosity if the same type of packing arrangement is attainable and if the wall effects2 can be neglected. The only difference that one will observe when a column is packed with particles of different sizes is that there are more interstitial pores in the case of smaller particles, but their total volume will be the same as the volume of the interstitial pores formed by packing large particles.

In the case of protein chromatography, particles are typically spherical, although fibrous and irregularly shaped particles are also commercially available. As already mentioned, particles used for preparative chromatography of proteins are porous and are characterized by a mean pore size and/or pore size distribution, and the intraparticle porosity, εp (the fraction of particle volume that is occupied by pores as shown in Fig. 16.4B). Recent advances in chromatography resin development for purification of large molecules showed that resins can also be classified according to the ligand density distribution. For instance, in the case of CaptoTM Core resin, the distribution is represented by a step change in ligand density along the particle radii (i.e., the outer shell contains no ligand, while the particle core is fully functionalized).

Some of the particulate stationary phases are made from more than one type of polymer/material, examples being solid core particles, particles with dispersed solid matter to increase the particle density, and the so-called composite resins in which pore walls are grafted with polymer chains of different lengths and properties. These chains can either provide active sites for adsorption by themselves, or be functionalized with specific ligands. The polymer chains provide additional surface area for adsorption and/or enable a three-dimensional ligand distribution, which can enhance protein partitioning into the pore. This partitioning phenomenon yields an increase in the resin binding capacities compared with the capacities attainable using the same base matrix, but without the surface extenders, where the adsorption only occurs on the pore surface.3 For instance, in the case of a cylindrical pore of 200 nm in diameter and a protein of 70 kDa in size, the increase in the binding capacity could be even up to 2.7-fold, depending on the type of surface extender used.

For non-composite resins, binding capacities depend on the specific surface area that is available for adsorption, which in turn depends on pore size distribution, particle porosity, and protein molecular size.

Some common liquid phases

Several hundred liquid stationary phases have been tested, and described in the literature. Except for special analytical problems, it is usually sufficient to try several basic stationary phases. A wide range of phases is manufactured under various commercial names.

Carbowaxes are polyethylene glycols. The number in the name gives the average molecular weight. For example, Carbowax 1500 and Carbowax 20M have relative molecular weights of 1300–1600, and 15000–20000. Carbowaxes 350, 550 and 770 are methoxypolyethylene glycols. A lower molecular weight generally indicates higher polarity.

Ucons are mixtures of polyethylene- and polypropylene-glycols. Ucon LB is not soluble in water. Ucons HB and 75-H are water soluble. The further numbers in the series refer to the phase viscosity.

Many types of polyesters are available. PEGA is polyethyleneglycol adipate, DEGA is polydiethyleneglycol adipate, DEGS is polydiethyleneglycol succinate. Stationary phases of the LAC series are analogous.

Phases in the Dexsil series have high thermal stability (to 400 °C); they are methylsiloxanes with various functional groups, mixed with polymethylcarboranes.

Squalane (hexamethyltetracosane) is a standard non-polar phase. The Apiezon series of mixtures of saturated long-chain hydrocarbons are also widely used.

Silicones are among the most commonly used liquid phases. The SE series are polysiloxanes with characteristic groups given by the number after SE. For example, SE-30 has methyl, SE-52 contains methyl and phenyl, and SE-33 contains methyl and vinyl groups. The DC silicone oil series consists of polysiloxanes substituted either by methyl (DC-200) or methylphenyl (DC-550 and DC-710) groups. Silicone oils of the OV series are widely used; these are methylpolysiloxane oils with various functional groups which increase their polarity.

1.4 CCC and HPLC

Figure 1.2 shows the operating range of the CCC technique. It does not overlap much with the HPLC operating range which is with VS/VM ratios lower than 0.1 (log VS/VM < –0.1) and high distribution coefficient differences (left part of Fig. 1.2). CCC operates with a significant stationary phase volume (Φ > 0.1). HPLC operates with a very low stationary phase volume (Φ < 0.1). This critical difference should be pointed out to understand that the two techniques are complementary.

1.4.1 Plate count working range

The difference in stationary phase volume between HPLC and CCC explains why the two techniques do not need the same number of plates to operate efficiently. Figure 1.2 was prepared theoretically using the same efficiency (2000 plates) for the two techniques. It should be pointed out that a 2000 plate efficiency is very poor for an LC column. Compared to CCC, HPLC compensates for the low VS/VM ratio with high efficiency. Figure 1.3 shows the number of theoretical plates needed to obtain a baseline resolution between Solutes 1 and 2. The KD1 value is 1; the KD2 value is shown in Fig. 1.3. The maximum efficiency obtained so far with a CCC column is below 5000 plates. This limits the range of use of the CCC technique as showed by the dotted area of Fig. 1.3. The phase ratio limits the LC range, as indicated by the hatched area of Fig. 1.3.

Fig. 1.3. The chromatographic efficiency, in number of plates, needed to obtain a baseline separation (Rs = 1.5) between a solute of distribution constant KD1 = 1 and a solute of distribution constant KD2 = 1.1, 1.2, 2 or 5. Dotted area: CCC working range; hatched area: HPLC working range. The open and closed squares and stars correspond to the chromatograms of Fig. 1.4.

Figure 1.4 illustrates the significance of Figs. 1.2 and 1.3, and shows four chromatograms. The complete sets of experimental values used to obtain the four chromatograms are listed in Table 1.1. The closed square LC chromatogram (Fig. 1.4) shows a poor separation of the two solutes. The separation is easily enhanced by an efficiency increase from 2000 to 7600 plates (open square). Figure 1.4 (bottom) shows the CCC chromatograms corresponding to Table 1.1 data. The two solutes are completely separated even with a 33% stationary phase retention (open star) which is a poor Sf value in CCC. A baseline separation would still be observed with only 200 plates (chromatograms not shown). The CCC technique can work efficiently with a surprisingly low plate count.

Fig. 1.4. Chromatograms corresponding to the data listed in Table 1.1. Top: HPLC (squares) improves resolution by increases in efficiency. Bottom: CCC (stars) improves resolution by increasing the stationary phase volume in the column.

Table 1.1. Chromatographic parameters used to prepare Fig. 1.4

Parameter	Symbol (Unit)	CCC	LC
Symbol in Fig. 1.4		*	⋆	▪	□
Column volume	VT (ml)	180	180	250	250
Stationary phase volume	Vs (ml)	160	60	8	8
Mobile phase volume	VM (ml)	20	120	100	100
Phase ratio	log VS/VM	0.9	−0.3	−1.1	−1.1
Partition coefficient: Solute 1	KD1	1	1	1	1
Partition coefficient: Solute 2	KD2	2	2	2	2
Retention factor: Solute 1	k′1	8	0.5	0.08	0.08
Retention factor: Solute 2	k′2	16	1	0.16	0.16
Retention volume: Solute 1	VR1 (ml)	180	180	108	108
Retention volume: Solute 1	VR2 (ml)	340	240	116	116
Efficiency	N (plates)	2000	2000	2000	7600
Resolution	Rs	6.4	2.6	0.4	1.5

The LC column is a 20 cm semi-prep column with 4 cm internal diameter.