Idea Transcript
3. THEORETICAL ANALYSIS 3.1 Bioprocesses Glutamate is the most abundant free amino acid in bacterial cytoplasm. In order to produce glutamate, glutamate producers must do two things well, which they must overproduce glutamate in excess of their normal metabolic needs, and they must excrete it into culture broth.
The precise
mechanism by which C. glutamicum still not completely understood despite over forty years of study. involved.
Some physiological traits, however, are clearly
These include biotin auxotrophy of producing strains, a marked
decrease in α- ketoglutarate dehydrogenase activity during production, and a predilection for exporting glutamate, perhaps via a specific transporter. Many
of
the
original
glutamate excreting strains
were
biotin
auxotrophs, and growing in biotin deficient medium was found to trigger glutamate production. Biotin is a cofactor used by enzymes those carboxylate substrates. One such enzyme is acetyl-CoA carboxylase that converts AcetylCoA and CO2 to Malonyl-CoA in the first step of fatty acid biosynthesis. Biotin auxotrophs growing in biotin deficient medium were proposed to have altered membranes due to suboptimal fatty acid biosynthesis. Supporting the notion of altered permeability is the observation that growth at higher temperatures, or including detergents or cell wall biosynthesis inhibitors like penicillin in the growth medium can also trigger excretion [Kinoshita, 1957]. Reduced levels of α-ketoglutarate dehydrogenase during production may also be linked to membrane integrity. In corynebacteria, the enzyme has three
activities on two peptides, the α-keto dehydrogenase, dihydrolipoamide Ssuccinyltransferase peptide and the dihydrolipoamide dehydrogenase peptide. The latter is shared with pyruvate dehydrogenase and is likely to be membrane bound and thus prone to being affected by trigger factors that alter membrane composition. Since α-ketoglutarate dehydrogenase catalyzes a step in the TCA cycle, the cycle is largely incomplete during glutamate production, a circumstance requiring that pools of oxaloacetate be filled, as carbon is lost through glutamate. Anaplerotic enzymes replace OAA, these enzymes include pyruvate carboxylase (Reaction 1), malic enzyme (Reaction 2), PEP carboxylase (Reaction 3) and glyoxylate pathway enzyme 1. Pyruvate + CO2 + ATP 2. Pyruvate + CO2 + NADPH 3. Phosphoenol pyruvate + CO2
oxaloacetate + ADP + Pi malate + NAD+ + H+ oxaloacetate + Pi
The actual contribution of each anaplerotic enzyme or pathway depends on the growth conditions, production phase and a host of interacting metabolic signals. However, mutants lacking pyruvate carboxylase produce as much glutamate as wild type suggesting that PEP carboxylase is the major anaplerotic enzyme when growing on sugars. When growing on acetate or fatty acids, the glyoxylate pathway assumes a major role in filling OAA pools, although acetate is not normally used in the industrial process. For many years, increased membrane permeability was thought to promote glutamate excretion.
Recently, however, export via a specific efflux
transporter has been proposed as a mechanism for avoiding exceptionally high
levels of intracellular glutamate. Such an exporter has been proposed by analogy with the Lysine transporter that is responsible for lysine and arginine export in C. glutamicum. Other
factors
contributing
to
glutamate
overproduction
metabolic flux alterations based on cell growth limitations.
include
The following
chain of events seems likely, A triggering mechanism (biotin depletion, high temperature, cell membrane alterations by detergents, oleic acid or antibiotics) results in a decrease or repression of α-keto dehydrogenase. The result is a redistribution of metabolites at the branch point in the TCA cycle leading from α-keto to succinyl-CoA or glutamate.
The increase in
glutamate levels, beyond that needed for cell growth, stimulates glutamate efflux as the cell attempts to maintain the proper level of intracellular glutamate.
The result is the excretion of glutamate from the cell [Kinoshita
et.al 1958]. 3.2 Bioreactors A bioreactor refers to any manufactured or engineered device or system that supports a biologically active environment. In one case, a bioreactor is a vessel
in
which
a chemical
process is
carried
out
which
involves microorganisms or biochemically active substances derived from the organisms. This process can either be aerobic or anaerobic. These bioreactors are commonly cylindrical, ranging in size from litres to cubic meters, and are often made of stainless.
A bioreactor may also refer to a device or system meant to grow cells or tissues in the context of cell culture. These devices are being developed for use in tissue engineering or biochemical engineering. Several types of bioreactors are used to produce different products from microorganisms. Most of the common types of bioreactors are batch, fed batch, and CSTR, packed bed, airlift and Immobilized reactors [Williams J.A. 2002].The batch reactors are usually in a cylindrical vessel equipped with a mechanical impeller, baffles and aeration at the bottom of the vessel by means of sparger. The most common type of aerobic reactor use in now days are batch bioreactor, which is having specific internal configuration to provide specific circulation pattern. For industrial applications, this unit offers manufacturers both low capital investments and lower operating costs. For laboratory experiments with smaller volumes, the mixing vessel is typically made of glass. Stainless steel tank construction is the standard for industrial applications involving larger volumes. The height to diameter ratio varies depending on the heat removal requirements.
3.2.1 Modes of operation On the basis of mode of operation, a bioreactor may be classified as batch, fed batch or
continuous (e.g. a continuous stirred-tank
model). An example of a continuous bioreactor is the chemostat.
reactor
Organisms growing in bioreactors may be suspended or immobilized. A simple method, where cells are immobilized, is a Petri dish with agar gel. Large scale immobilized cell bioreactors are:
moving media, also known as Moving Bed Biofilm Reactor (MBBR)
packed bed
fibrous bed
Membrane Batch reactors are simplest type of mode of reactor operation. In this
mode, the reactor is filled with medium and the fermentation is allowed to proceed. When the fermentation has finished the contents are emptied for downstream processing. The reactor is then cleaned, re filled, re inoculated and the fermentation process starts again. In continuous reactors, fresh media is continuously added and bioreactor fluid is continuously removed. As a result, cells continuously receive fresh medium and products and waste products and cells are continuously removed for processing. The reactor can thus be operated for long periods of time without having to be shut down. Continuous reactors can be many times more productive than batch reactors. This is partly due to the fact that the reactor does not have to be shut down as regularly and also due to the fact that the growth rate of the bacteria in the reactor can be more easily controlled and optimized. In addition, cells can also be immobilized in continuous reactors, to prevent their removal and thus further increase the productivity of these reactors.
The fed batch reactor is the most common type of reactor used in industry. In this reactor, fresh media is continuous or sometimes periodically added to the bioreactor but unlike a continuous reactor, there is no continuous removal. The fermenter is emptied or partially emptied when reactor is full or fermentation is finished. As with the continuous reactor, it is possible to achieve high productivities due to the fact that the growth rate of the cells can be optimized by con trolling the flow rate of the feed entering the reactor. [Amin .G.A et.al., 2007]. 3.2.2 Submerged Bioreactors (Batch Reactor) The Batch reactor is the generic term for a type of vessel widely used in the process industries. In a batch reactor the reactants and the catalyst are placed in the reactor and the reaction is allowed to proceed for a given time whereupon the mixture of unreacted material together with the products is withdrawn. Provision for mixing may be required. In an ideal batch reactor, the concentration and temperature are assumed to be spatially uniform. In practice, the condition can be approximately realized by vigorous agitation or stirring. All the elements of the fluid spend the same amount of time in the reactor, and hence have the same residence time. From the viewpoint of thermodynamics, a batch reactor represents a closed system. The steady states of the batch reactor correspond to states of reaction equilibria. Batch reactors are simple and needs little supporting equipment, and are therefore ideal for small scale experimental studies on reaction kinetics. Industrially it is used when relatively small amounts of material are to treat and when the product demand varies. Batch reactors are often used in the
pharmaceutical industry, where small volumes of high value products are made. A typical batch reactor consists of a tank with an agitator and integral heating or cooling system. Liquids and solids are usually charged via connections in the top cover of the reactor. Vapors and gases also discharge through connections in the top. Liquids are usually discharged out of the bottom. The advantages of a batch reactor lie in its versatility. A single vessel can carry out a sequence of different operations without the need to break containment.
This is particularly useful when processing, toxic or highly
potent compounds [Octave Levenspiel, 1999]. Material balance Rate of input - rate of output + rate of disappearance = rate of accumulation 3.3 Theory of immobilization Immobilization of microbial cells in biological processes can occur either as a natural phenomenon or through artificial process. While the attached cells in natural habitat exhibit significant growth, the artificially immobilized cells are allowed restricted growth. Since the time first reports of successful application of immobilized cells in industrial applications, several research groups world over have attempted whole cell immobilization as a viable alternative to conventional microbial fermentations. Various immobilization protocols and numerous carrier materials were tried. The cell immobilization process has also triggered our interest in bioreactor design. Using immobilized cells, different bioreactor configurations were reported with variable success. The study on the physiology of immobilized cells and development of
noninvasive
measuring
techniques
have
remarkably
improved
our
understanding on microbial metabolism under immobilized state. The industrial biotechnology processes using microorganisms are generally based on the exploitation of the cells in the fermentation medium during the process. The classical fermentations suffer from various constrains such as low cell density, nutritional limitations, and batch-mode operations with high down times. It has been well recognized that the microbial cell density is of prime importance to attain higher volumetric productivities. The continuous fermentations with free-cells and cell recycle options aim to enhance the cell population inside the fermenter. However, the free cell systems cannot operate under chemo static mode that decouples specific growth rate and dilution rates. During the last 20–25 years, the cell immobilization technology, with its origins in enzyme immobilization, has attracted the attention of several research groups. This novel process eliminates most of the constrains faced with the free cell systems. The remarkable advantage of this new system is the freedom it has to determine the cell density prior to fermentation. It also facilitates operation of microbial fermentation on continuous mode without cell washout. The whole cell immobilization process decouples microbial growth from cellular synthesis of favored compounds. Since the early 70s, when Chibata’s group announced successful operation of continuous fermentation of L-aspartic acid, numerous research
groups
have
attempted
various
microbial
fermentations
with
immobilized cells [Dsouza.S.F, 1989]. There are also specialized monograms and conference proceedings pertaining to cell immobilization technology, which have excited microbiologists and bioengineers.
The use of immobilized whole microbial cells and/or organelles eliminates the often tedious, time consuming, and expensive steps involved in isolation and purification of intracellular enzymes. It also tends to enhance the stability of the enzyme by retaining its natural catalytic surroundings during immobilization and subsequent continuous operation. The ease of conversion of batch processes into a continuous mode and maintenance of high cell density without washout conditions even at very high dilution rates, are few of the many advantages of immobilized cell systems. The metabolically active cell immobilization is particularly preferred where co-factors are necessary for the catalytic reactions. Since co factor regeneration machinery is an integral function of the cell, its external supply is uneconomical. There is considerable evidence to indicate that the bound cell systems are far more tolerant to perturbations in the reaction environment and similarly less susceptible to toxic substances present in the liquid medium. The recent reports on higher retention of plasmid bearing cells have further extended the scope of whole cell immobilization
to
recombinant
product
formation.
Another
important
advantage of immobilization, particularly in the case of plant cells, is the stimulation of secondary metabolite formation and elevated excretion of intracellular metabolites. 3.4 Immobilization methods Many methods namely adsorption, covalent bonding, cross linking, entrapment, and encapsulation are widely used for immobilization. These categories are commonly used in immobilized enzyme technology. However, due to the completely different size and environmental parameters of the cells, the relative importance of these methods is considerably different. The criteria
imposed for cell immobilization technique usually determine the nature of the application 3.4.1 Adsorption This was apparently the first example of cell immobilization. Hattori and Furusaka reported the binding of Escherichia coli cells on to an ion exchange resin. Subsequently, a variety of microbial cells were immobilized by adsorption on different supports like kieselguhr, wood, glass ceramic, plastic materials, etc. [Klein and Ziehr 1990] have reviewed the immobilization of microbial cells by adsorption. Since the adsorption phenomenon is based on electrostatic interactions (vander Walls forces) between the charged support and microbial cell, the actual zeta potential on both of them plays a significant role in cell support interactions. Unfortunately, the actual charge on support surfaces is still unknown and this limits the proper choice for microbial attachment. Along with charge on the cell surface, the composition of cell wall carrier composition will also play a predominant role [Kolot.F.B, 1981]. Cells of Saccharomyces cerevisiae and Candida utilis contain a mannans in the cell wall. The cells of latter have a strong affinity to Cancanavalin-Aactivated carrier [Horisberger.M, 1976]. Carrier properties, other than zeta potential, will also greatly influence cell support interaction. All glasses or ceramic supports are comprised of varying proportions of oxides of alumina, silica, magnesium, zirconium, etc. which result in bond formation between the cell and the support. Several procedures of cell adsorption based on pH dependence are reported [Klein and Ziehr, 1990].
3.4.2 Covalent bonding The mechanism involved in this method is based on covalent bond formation between activated inorganic support and cell in the presence of a binding (cross linking) agent. For covalent linking, chemical modification of the surface is necessary. Cells of S. cerevisiae were immobilized by coupling silinized silica beads [Novarro et.al, 1977]. The reaction requires introduction of reactive organic group on inorganic silica surface for the reaction between the activated support material and yeast cells. A amino propyl triethoxy silane is generally used as the coupling agent. This inorganic functional group condenses with hydroxyl group on silica surface. As a result, the organic group is available for covalent bond formation on the surface of silica. Covalent bonding can also be achieved by treating the silica surface with glutaraldehyde and isocyanate. A system of more general interest has been developed by Kennedy and Cabral [Kennedy and Cabral, 1985], using inorganic carrier system. The addition of Ti4+ or Zr4+ chloride salts to water results in pHdependent formation of gelatinous polymeric metal hydroxide precipitates wherein the metals are bridged by hydroxyl or oxide groups. By conducting such a precipitation in a suspension of microbial cells, the cells have been entrapped in the gel-like precipitate formed. In continuous operation, titanium hydroxide-immobilized cells of Acetobacter were employed to convert alcohol to acetic acid. 3.4.3 Cross linking
Microbial cells can be immobilized by cross-linking each other with bi- or multifunctional reagents such as glutaraldehyde [Chibata et.al, 1974], toluenediisocyanate [Lartigue .D et.al 1976]. The toxicity of the chemicals used for cross linking obviously imposes limitations for the general applicability of these procedures. Apart from chemical cross-linking, procedures employing physical processes, such as flocculation and pelletization, also benefit the immobilization techniques because of strong mutual adherence forces of some microbial cell cultures. 3.4.4. Entrapment The most extensively studied method in cell immobilization is the entrapment of microbial cells in polymer matrices. The matrices used are agar, alginate, carrageenan, cellulose and its derivatives, collagen, gelatin, epoxy resin, photo cross linkable resins, polyacrylamide, polyester, polystyrene and polyurethane. Among the above matrices, polyacrylamide has been widely used by several workers. This gel was first used for immobilization of enzymes [Bernfeld.P et.al., 1963]. Later this technique was successfully applied to immobilization of lichen cells [Mosbach.K et.al., 1966]. As a rule, the entrapment methods are based on the inclusion of cells within a rigid network to prevent the cells from diffusing into surrounding medium while still allowing penetration of substrate. The precise mode of entrapment of cells in polyacrylamide is critical for satisfactory retention of activity. The factors affecting the gel preparation are the content of acrylamide, the ratio of cells to acrylamide, and the size of the gel particles. While the former two influence the hardness of the resulting gel and the pore size of the microlattice in which the cells are entrapped, the third factor determines the activity, stability, and the
pressure drop when packed in a reactor. The other procedures for network formation for cell entrapment are precipitation, ion exchange gelation, and polymerization. The precipitation techniques are exemplified by collagen [Veith W.R et.al.,1974]. By changing a parameter such as temperature, salinity, pH, etc. polymer precipitate can be prepared from a homogeneous solution of linear chain polymers. The networks are primarily formed by precipitation with salt solutions, ones that are constituted by secondary valency forces ranging from dispersion to hydrogen bonding. Network formation procedures where the cross linking is established by ionic bonds between linear polyelectrolytes and multivalent cations have been extensively tried. Entrapment of cells in alginate gel is popular because of the requirement for mild conditions and the simplicity of the used procedure. Several reports are available employing alginate gel [Keirstan .M et.al.,1977]. K-carrageenan is one of the earliest gel materials used for cell immobilization for continuous production of l-lactic acid by Escherichia coli [Nishida.Y et.al., 1979]. The immobilization procedure is similar to alginate, and several other groups have used this polysaccharide as a preferred gel matrix either alone or in combination with other gums because of the mild conditions required good gel stability. Using k-carrageenan,
[Takata et al.1983] reported that the
immobilized Brevibacterium flavum attained high stability against several denaturing chemicals. The rate of cell leakage could be lowered by hardening the gel with potassium cations. Similarly several other natural polymers such as agar, agarose, pectin and gelatin were also employed for cell immobilization. [Castillo et al.,1991] employed gel as a carrier material for the immobilization
of Kluyveromyces fragilis for B-galactadase activity and E. coli for penicillin acylase. The authors produced fibres instead of beads by direct extrusion in cold water. The fibers were further strengthened by treatment with 1.25% glutaraldehyde solution for 45 min. The reversible network formed was affected by certain calcium chelating agents like phosphates, Mg2+, K+ and EDTA and the gel integrity was poor. In ethanol fermentation, where large quantities of CO2 are produced, the gel spheres disintegrate due to CO2 pressure in the bed. Many variations in the immobilization procedure were tried. [Chotani et al., 1991] reported that incorporation of micron-sized silica in the bead improved the mechanical strength and internal space adhesion. Gel particles of high strength, elasticity, porosity, and high cell content can be prepared by the above procedure. The network is irreversibly formed and inert. [Lorenz et al.,1987] tried a new immobilization procedure using polyurethane ionomers. By coagulation with a salt solution, the mixture of cells and dispersion of amphiphillic polyurethane ionomers from hydrogel under the entrapment of the biocatalyst. 3.5 Emerging trends Whole cell immobilization as a tool to intensify microbiological processes has been well established. Several examples of production of a variety of biochemicals by immobilized cells have been successfully demonstrated. Though initially our knowledge on physiology of immobilized cells was limited and hypothetical, the use of microelectrodes and development of noninvasive techniques to study the immobilized cells under microenvironment have revealed significant information pertaining to metabolic structural alterations
occurring in the cell under immobilized phase. One of the difficulties experienced to evaluate various carriers, process conditions, and operating conditions is the non uniformity of reporting the information. For example, the volumetric productivity of the bioreactor system with immobilized cells can be determined by considering total volume of the reactor, or the void volume. But researchers calculate and report either one of them, which is sometimes misleading. It is necessary to evolve a common protocol to assess the performance of a given system. Though a variety of carrier materials have been tried, there are very few reports comparing these interms of their performance, long term stability, and cost. The observations made with immobilized cells and altered morphology indicates the influence of anchorage on cell metabolism. Perhaps this may lead to a separate study of solid state fermentation, which can be considered as microbial proliferation on solid surfaces, and its influence on bioprocess acceleration in some cases. An important area of research requiring greater focus is the bioreactor design and its long term operation. Except for a couple of experimental ventures, most of the experiments have been carried out on a very small scale, and hence very difficult to scale up. The future research should centre around not only for developing feasible microbiological processes with immobilized cells but also for carrying out extensive research in bioreacter design to solve some of the engineering problems, specially the ones that are connected with diffusional limitations. It is important to generate adequate data with larger systems for longer times to enable the design engineers to
translate these results into commercial realities. It is also very important that future research should focus on microbial physiology under immobilized state to enrich our knowledge on process intensification. The recent reports on enhanced plasmid stability of genetically engineered microorganisms [Nasri et.al.,1987] under immobilized conditions, and the viability of microbial cells over a period of 18 months[Romo et.al.,1997] under entrapped conditions, are few of the many potential new applications of immobilized cells. 3.6 Response Surface Methodology (RSM) Response surface methodology (RSM) was used for designing the various experiments during optimization.RSM is a set of techniques that encompasses
Designing a set of experiments (or) setting up a series of experiments that will yield adequate and reliable measurements of the response interest.
Determining a mathematical model that best fits the data collected from the design choosing, by conducting appropriate test of hypothesis concerning the model parameters.
Determining the optimal setup of the experimental factors that produce the maximum and minimum value of the response.
If discovering the best value of response is beyond the available resources of the experiment, then the response surface methods on aimed at obtaining at least a better understanding of the overall system. When the behavior of the measured response of interest is governed by certain loss which leads to a deterministic relationship between the response and the set of experimental factors chosen, it should then be possible to determine the best
values of the factors to optimize a desired output. However, quiet often an empirical approach is necessary, because the relationship is either too complex or unknown. In any system in which variable quantity is change, the interest might be in assessing the effect of factors on the behavior of some measurable quantity, such an assessment is possible through regression analysis. Using the data collected from a set of experimental rights, regression helps, establish empirically by fitting some form of mathematical model, the type of relationship that is present between the response variables and its influencing factors. The response variable is the dependent variable and is called the Response, and the levels of the influencing factors are called explanatory, regression, or input variables. Regression analysis is one of the most widely used tools for investigating cause and effect relationships having applications in the physical, biological and social sciences, as well as in engineering. Response surface methods are additional techniques employed before, while, and after a regression analysis is performed on the data. Preceding the regression analysis, the experiment must be designed, that is input variables must be selected their values during the actual experimentation designated. After regressing analysis is performed, certain model testing procedures and optimization techniques are applied. Thus, the subject of RSM includes the application of regression as well as other techniques in an attempt to gain a better understanding of the characteristics of the response system under study. Central Composed design was used to optimize the fermentation parameters. To optimize the range of experimentation for the 24-full factorial
Central composite design was used. The experiments were designed by using Satistica 6.0 software. 3.6.1Central Composite Design (CCD) Box-Wilson Central Composite Designs [Box G.E.P and Wilson K.B 1951] start with a factorial or fractional factorial design (with center points) and add "star" points to estimate curvature.
If the distance from the center of the
design space to a factorial point is ±1 unit for each factor, the distance from the center of the design space to a star point is ± value of
with | | > 1. The precise
depends on certain properties desired for the design and on the
number of factors involved. A central composite design diagram is generated below for two factors in fig.3.6.1
Fig. 3.6.1 Generation of a CCD for two factors A CCD always contains twice as many star points (2k) as there are factors (k) in the design. The star points represent new extreme values (low and high) for each factor in the design.
The three types of CCD designs depend on the position of star points. They are (i)
Circumscribed (CCC)
(ii)
Inscribed (CCI)
(iii)
Face Centered (CCF)
Pictorial representation of the star points in three types of CCD designs for two factors are shown in fig.3.6.2.
Fig. 3.6.2 Comparison of the three types of central composite designs The CCC explores the largest process space and the CCI explores the smallest process space. Both the CCC and CCI are rotatable designs, but the CCF is not.
The design points describe a circle circumscribed about the
factorial square in CCC design, for three factors, the CCC design points
describe a sphere around the factorial cube.
The value of
is chosen to
maintain rotatability and depends on number of experimental runs in the factorial portion of the central composite design:
number of factorial funs
1
4
If the factorial is a full factorial, then 2 k
1
…………… 3.1 4
These types of experimental design are frequently used together with response models of the second order.
The design consists of three types of
points and shown in fig 3.6.3 Axial points: 2n axial points are created by a screening analysis. Cube points: 2n cube points come from a full factorial design. Center point: A single point in the center is created by a nominal design. The axial points are located on a hyper-cube with the radius bi. The cube built by the cube points has side-length of 2bi.
Fig. 3.6.3 The points of a CCD design with three input parameters All three types of CCDs have the same structure but varying values for ai and bi. These values are listed in table-3.6.1 below. Table-3.6.1 : Factors a and b for CCD with full factorial cube
points
bi
rangei
ai
CCC
n
2 ai
CCI CCF 3.6.2 Factorial designs
n 1
2 bi 2 ai = 2 bi
In general, factorial designs are most efficient to study the effects of two or more factors.
A factorial design means, all possible combinations of the
levels of the factors are investigated in each complete trial or replication of the experiment. The effect of a factor is defined to be the change in the response produced by a change in the level of the factor.
A two factor factorial
experiment with design factors at two levels could be given as an example. The levels are called ‘‘low’’ and ‘‘high’’ and are denoted as ‘‘-’’ and ‘‘+’’, respectively, as shown in fig.3.6.4 (a).
-
+
(Low)
(High) Factor A (a) Fig. 3.6.4
-
+
(Low)
(High) Factor A (b)
Two-factor factorial experiments
The main effect of factor A in this two-level design can be thought of as the difference between the average response at the low level of A and the average response at the high level of A. Numerically: A
effect 2
40 52 20 30 2 2 …………… 3.2 A
effect = 21
This means, increasing factor A from the low level to the high level causes an average response increase of 21 units. Similarly, the main effect of B is B
effect 2
30 52 20 40 2 2 …………… 3.3
B
effect = 11
In some experiments, it can be found that the difference in response between the levels of one factor is not the same at all levels of the other factors. When this occurs, there is an interaction between the factors. The two-factor factorial experiment shown in fig.3.6.4 is considered for example. At the low level of factor B, the effect of A is Aeffect(low)=50-20=30 and at the high level of factor B, the effect of A is Aeffect(high)=12-40=-28. Since the effect of A depends on the level chosen for factor B, there is an interaction between A and B. 28 30 B int eraction 2 2 B interaction = –29 Clearly, the interaction is large in this experiment. Fig.3.6.4 (a) plots the response data in fig.3.6.4 (b) against factor A for both levels of factor B. The Band B+ lines are approximately parallel, indicating lack of interaction between factors A and B. Fig.3.6.5shows the factorial experiments once without and once with interactions between the factors. In fig.3.6.5 (b), B- and B+ lines are not parallel indicating an interaction between factors A and B. However, their interpretation is subjective and their appearance is often misleading.
(-)
Factor
(+)
(-)
Factor
A
A
(a)
(b)
(+)
Fig. 3.6.5 Factorial experiment a) without, b) with interaction There is another way of illustrating the concept of interaction. A regression model representation of the two-factor factorial experiment could be written as Response, y = β0 + β1x1 + β2x2 + β12x1x2 + ε
…………… 3.4
βs are parameters whose values are determined, x1 is a variable that represents factor A, x2 is a variable that represents factor B and ε is a random error term. The variables x1 and x2 are defined on a coded scale from -1 to 1 and x1x2 represents the interaction between x1 and x2. Generally, when interaction is large, the corresponding main effects have little practical meaning. For example, the main effect of A to be small in fig.3.6.4 (b). A
effect 2 A
50 12 20 40 2 2
effect = 1
The conclusion is that there is no effect due to factor A.
However,
examining the effects of A at different levels of factor B, A has an effect, but it depends on the level of B. That is, knowledge of the AB interaction is more useful than knowledge of the main effect. A significant interaction will often mask the significance of main effects. Factorial designs have several advantages. They are more efficient than one-at-a-time experiments. Furthermore, a factorial design is necessary to avoid misleading conclusions. Factorial designs allow the effects of a factor to be estimated at several levels of the other factors, yielding conclusions that are valid over a range of experimental conditions. As the number of factors in a 2k factorial design increase, the number of runs required for a complete replicate of the design rapidly grows. For example, a complete replicate of the 26 design requires 64 runs.
In this design, only 6 of the 63 degrees of freedom
correspond to the main effects and only 15 degrees of freedom correspond to two-factor interactions. The remaining 42 degrees of freedom are associated with three-factor and higher interactions. If the experimenter can reasonably assume that certain high-order interactions are negligible, then information on the main effects and low order interactions may be obtained by running only a fraction of the complete factorial experiment.