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Theory of differential centrifugation in angle-head rotors
ANALYTICAL
31, 279-285
BIOCHEMISTRY
Theory
of
(1969)
Differential
Centrifugation
Angle-Head
Rotors1
VERNE SCHUMAKER Department University
of Chemistry of California,
in
AND
ALLAN REES
and Molecular Los Angeles,
Received April
Biology Institute, California 90024
8, 1969
Differential centrifugation is the term used to describe a widely employed technique for the isolation and purification of virus in the preparative centrifuge. The crude homogenate is first cleared of cellular debris and faster contaminants by a low-speed centrifuge run, and then the virus is brought down at high speed. The virus pellet is then resuspended, and the low- and high-speed centrifugations are repeated. A combination consisting of a low-speed centrifugation followed by a high-speed centrifugation is called a cycle, and a number of cycles is called “differential centrifugation.“2 Stanley and Wycoff (2) were probably the first workers to employ differential centrifugation for the isolation and purification of virus. Other plant and animal virologists quickly adopted the technique because of the excellent yields, concentrations, purity, and infectivity of the recovered viral material. Many of the procedures subsequently developed for the isolation of different kinds of viruses are described in detail in review articles by Sharp (3) and by Steere (4). The angle-head rotor is usually employed in differential centrifugation because of its large capacity. An attempt to work out a theory adequately describing the movement of particles in the anglehead rotor was made by Pickels (5). A large fraction of the total mass transport was assumed to occur by means of a convecting disturbance down the sides of the angled centrifuge tube. The experi1 Supported in part by United States Public Health Service Grant GM-13914. Contribution No. 2340 from the Department of Chemistry, University of California, Los Angeles, California 90024. 2 Differential centrifugation should not be confused with differential sedimentation, a technique developed by Richards and Schachman (1) for accurate assessment of small differences in sedimentation coefficient in the analytical ultracentrifuge. 279
280
SCHUMAKER
AND REES
mental data which Pickels presented in the same paper, however, did not provide conclusive support for this convection theory. Recently, Anderson (6) developed another theory which may supplant the convective theory of Pickels. The new theory suggests, and experiments (6,‘7) seem to indicate, that identical sedimentation coefficients are obtained either by angle-head centrifugation or by analytical centrifugation. In developing the mathematics of differential centrifugation in the present paper, however, it really does not make much difference whether Pickels or Anderson is right, for a logarithmic scale is used to present the data, and even a factor of two difference would shift the curves only a little and not much affect their shapes. For making the computations, however, we will adopt the approach of Anderson because of its great simplicity and also because it now appears to have better experimental support. Figure 1 shows an angle-head centrifuge tube in place, as well as the nomenclature that will be used in describing radial positions. The meniscus position is located at a, and the tube bottom is located at b. Anderson’s approach is to assume that sedimentation is strictly radial, but that material which sediments into the side walls is resuspended by Corioli forces acting to swirl the liquid in a vertical plane. Hence the position of the migrating boundary, r, is given by the simple relation T = a exp[s(&dt] (1) where s is the sedimentation coefficient of the macromolecules, o is the angular velocity, t is time, and the limits of integration cover the duration of the run.
FIG. 1. Cross-sectional drawing of centrifuge tube in angle-head rotor. Distances from axis of rotation to meniscus (a), tube bottom (b) , and any arbitrary point (T) are illustrated by arrows.
DIFFERENTIAL
CENTRIFUGATION
IN ANGLE-HEAD
DEVELOPMENT OF THE MATHEMATICS DIFFERENTIAL CENTRIFUGATION
ROTORS
281
OF
Two quantities essential for the development of the theory are q, the fraction of macromolecules remaining in solution during a centrifuge run, and p, the fraction pelleted. Since a given macromolecule must be either in the supernatent solution or else in the pellet, it follows that p + q = 1. If the sedimenting boundary has moved to a position T, then the fraction of material remaining in solution is given as :
Equation 2 is only approximately correct because the diluting effect of a centrifugal field is ignored, and because no provision is made for the hemispherical tube bottom. For our purposes, however, it is accurate enough. Values of r to be used in Equation 2 may be computed from equation 1. In order to use equation 1, s must be known. The usual conversion formula is employed to compute values of s from values of the standard sedimentation coefficient, s: (1 - VP) 0.01002 (3) 7) s = s20sw( l- ii 0.9982) where V, p, and 7, are partial specific volume of macromolecule, density of solution, and viscosity of the solution, respectively. The quantities 0.9982 and 0.01002 are the density (gm/cc) and viscosity (poise) of water at 20°C. A series of values for q, and hence p, may be computed as a function of spD,W for any particular solvent, rotor, speed and time. This is accomplished through the use of equations 3, 1, and 2, in that order. Negative values of q should be replaced by 0. Next, it is convenient to plot this series of values for p or q as a function of log szo,,,,.In Figure 2, for example, we show the results of a plot of p for a centrifuge run at 19,000 rpm, a = 4.4, and b = 8.9 cm, at 5” in a solvent having the viscosity and density of water at that temperature. The run is of 2 hr duration. A plot of p vs. .se,,,w, as drawn in Figure 2, graphically portrays that fraction of material found in the pellet at each particular sedimentation value. The construction of such a curve is desirable for the high-speed centrifuge run in which the virus is to be found in the pellet. On the other hand, for a low-speed run in which the pellet contains mostly debris and is to be discarded, while the supernatant
282
SCHUMAKER
AND REES
FIG. 2. Plot of fraction of material pelleted, p, as function of sedimentation coefficient, 820,~. Equations 3, 1, and 2 were employed, in that order, to calculate q, and hence p, for each value of ~20,~. Conditions for the run: 2 hr at 19,000 rpm, a = 4.4 cm, b = 8.9 cm, at 5°C in a solvent having the viscosity and density of water.
contains the virus and is saved, a plot of Q vs. szo,,+. gives the fraction of material at each particular sedimentation coefficient that is retained in the supernatant fluid. Let us distinguish values of p and q for high- and low-speed runs by the use of the prime (‘). Thus, let q’ be the value of q for a lowspeed centrifuge run, but retain the symbol of p as the values of p for a high-speed centrifuge run. The product of these two quantities, p$, then yields that fraction of material retained in an entire cycle composed of a high-speed run together with a low-speed run. We may immediately extend this treatment to include a number of cycles. A particularly simple expression results if the centrifuge run conditions for each of n cycles is identical. In this case, the product is simply raised to the n power: (pq’)“. It yields the fraction of material of each sedimentation coefficient recovered at the end of the experiments. A plot of this quantity is shown in Figure 3 for 1,3, and 5 cycles of differential centrifugation. The goodness or poorness of the isolation procedure for a given set of run conditions, such as rotor speed, time, etc., becomes immediately evident from an inspection of this type of plot (Fig. 3). For example, it is particularly evident that, for 3 and 5 cycles, the “high-speed cutoff” between 30 and 50 S is reasonably sharp. On the other hand, the “low-speed cut-off” extends from about 200 to 10,000 S. This result is typical of those found for alternate high- and low-speed runs. The “high-speed cut-off” increases in sharpness as the num-
DIFFERENTIAL
CENTRIFUGATION
IN ANGLE-HEAD
ROTORS
283
0 FIG. 3. Plot of fraction recovered, (pq’) n, plotted as function of SZO,Wfor n = 1 (solid line), 3 (dashed line), and 5 (dotted line) cycles of centrifugation. Conditions for the run: 120 min at 40,000 rpm at 5°C to form the pellet. The virus was then washed and clarified at 6,000 rpm for 30 min. Rotor dimensions are a = 4.8 cm, b = 8.0 cm. The solvent is assumed to have the viscosity and density of water.
ber of cycles is increased, while just the opposite situation occurs at the other end of the curve. We would like to point out, however, that by a simple change in the differential centrifugation procedure it is possible to eliminate the broad spread at the “low-speed cut-off,” and to make both flanks of the curve equally sharp. In order to do this, it is necessary to choose a much higher speed for the “low-speed” run. In this case, the initial “low-speed” pellet will contain some virus as well as debris. Therefore, it is necessary to save both the pellet and the supernatant fluid. The pellet is resuspended in about l/s of the original volume or less of fresh solvent, and again centrifuged at the same “low speed.” The supernatant fluid is saved, and the second pellet is again resuspended and centrifuged. The original supernatant fluid is now recombined with that obtained in these two washes. The mathematics of the recovery process is simple. We may write that the first “low-speed” pellet contained a fraction, p’, of any component, The second pellet, having been washed once, contained only P’~, and the third pellet, after being washed a second time, contained only P’~. Thus, when the washes are recombined with the original supernatant fluid, the amount of each component retained in the recombined supernatant fluids must be (l-~‘~). The combined supernatant fluids are then centrifuged at a high speed to
284
SCHUMAKER
AND REES
pellet the virus. This high-speed pellet is resuspended and twice again repelleted at this same high speed. The fraction of each component conserved in the final pellet from the three high-speed runs will be p3. The total recovery from both the high and low speed runs will be p3 (1 -p’“) . In Figure 4 is shown the resulting plot for the run conditions listed in the legend. Both flanks of the curve are now sharp. Therefore, this modified technique should yield greater amounts of more effectively purified material,
FIG. 4. Plot of fraction recovered, p3 (1 - p’) 3, plotted as function of SZO,~, for 3 cycles of centrifugation. A much sharper “low-speed cut-off” is obtained by this multiple washing procedure. Most virus will be isolated between 100 and 2000 Svedberg units, a region that can be obtained in good yield by appropriate selection of rotor speed, time, etc. Conditions chosen for the illustration: 13,500 rpm at 30 min for low-speed runs and 38,000 rpm at 120 min for highspeed runs, in a rotor with dimensions a = 3.8 cm and b = 8.1 cm. DISCUSSION
Quite apart from the theory presented here, the conditions chosen for the initial cycle may be selected to first provide extensive clarification of the solution to reduce the viscosity of the medium and then to concentrate the virus from an initially huge volume. After this preliminary preparation has been accomplished, however, repeated cycling is thought to yield purer and purer virus. The major advantage of repeated cycling appears to come from sharpening the flank of the “high-speed cut-off” side of the recovery plot, as shown in Figure 3. Unfortunately, this procedure will cause a broadening of the “low-speed” flank.
DIFFERENTIAL
CENTRIFUGATION
IN
ANGLE-HEAD
ROTORS
285
It is predicted by the theory presented in this communication that a much better recovery is to be expected by choosing somewhat higher “low-speed” run conditions, and then repeatedly washing the pellet which is obtained. In this manner, both a sharp “high-speed” and a sharp “low-speed” flank to the curve will be obtained, as is shown in Figure 4. If soft pellets are obtained which are readily restirred into the solution, then the best technique to employ is to withdraw the solution gently by pipet to a point considerably above the pellet. Now simply by increasing the number of cycles of centrifugation, the desired degree of purification is readily obtained, without the danger of disastrous loss of material. By suitable change in equation 3, this case can be handled mathematically. The mathematics of differential centrifugation as described in this communication are, at best, only a guide, for unexpected sedimentation behavior of macromolecules often occurs during the purification process. For example, sometimes virus will aggregate when purified of low molecular weight proteins (8). Nevertheless, it is hoped that the theory may serve useful purposes in the initial selection of optimal conditions for virus isolation, and also for correct interpretation of unusual results when they do arise. SUMMARY
Differential centrifugation is much used for the isolation and purification of virus. Appropriate conditions of rotor speed, time, number of cycles, etc. are usually chosen empirically. Simple equations are presented here for the prediction of the amount of material that will be recovered in each sedimentation class. These equations suggest choices of centrifugation conditions for improved yield and purity, REFERENCES 1.
RICHARDS,
E.
G.,
AND
SCHACHMAN,
H.
K.
(1957),
Am.
Chem. Sot. 79,
5324. 2. 3. 4.
STANLEY, W. M., AND WYCOFF, R. W. G. (1937)) Science 85, 181. SHARP, D. G. (1953)) in “Advances in Virus Research” (K. M. Smith M. A. Lauffer, eds.), Vol. I, 277. Academic Press, New York. STEERE, R. L. (1959), in “Advances in Virus Research” (K. M. Smith M. A. Lauffer, eds.) , Vol. VI, 3. Academic Press, New York. PICKELS, E. G. (1943)) J. Gen. Physiol. 26, 341. ANDERSON, N. G. (1968), Anal. Biochem. 23, 72. CHARLWOOD, P. A. (1963), Anal. Biochem. 5, 226. BRAKKE, M., AND STAPLES, R. (1958), ViTology 6, 14.
p,
p.
5.
6. 7. 8.
and and
31, 279-285
BIOCHEMISTRY
Theory
of
(1969)
Differential
Centrifugation
Angle-Head
Rotors1
VERNE SCHUMAKER Department University
of Chemistry of California,
in
AND
ALLAN REES
and Molecular Los Angeles,
Received April
Biology Institute, California 90024
8, 1969
Differential centrifugation is the term used to describe a widely employed technique for the isolation and purification of virus in the preparative centrifuge. The crude homogenate is first cleared of cellular debris and faster contaminants by a low-speed centrifuge run, and then the virus is brought down at high speed. The virus pellet is then resuspended, and the low- and high-speed centrifugations are repeated. A combination consisting of a low-speed centrifugation followed by a high-speed centrifugation is called a cycle, and a number of cycles is called “differential centrifugation.“2 Stanley and Wycoff (2) were probably the first workers to employ differential centrifugation for the isolation and purification of virus. Other plant and animal virologists quickly adopted the technique because of the excellent yields, concentrations, purity, and infectivity of the recovered viral material. Many of the procedures subsequently developed for the isolation of different kinds of viruses are described in detail in review articles by Sharp (3) and by Steere (4). The angle-head rotor is usually employed in differential centrifugation because of its large capacity. An attempt to work out a theory adequately describing the movement of particles in the anglehead rotor was made by Pickels (5). A large fraction of the total mass transport was assumed to occur by means of a convecting disturbance down the sides of the angled centrifuge tube. The experi1 Supported in part by United States Public Health Service Grant GM-13914. Contribution No. 2340 from the Department of Chemistry, University of California, Los Angeles, California 90024. 2 Differential centrifugation should not be confused with differential sedimentation, a technique developed by Richards and Schachman (1) for accurate assessment of small differences in sedimentation coefficient in the analytical ultracentrifuge. 279
280
SCHUMAKER
AND REES
mental data which Pickels presented in the same paper, however, did not provide conclusive support for this convection theory. Recently, Anderson (6) developed another theory which may supplant the convective theory of Pickels. The new theory suggests, and experiments (6,‘7) seem to indicate, that identical sedimentation coefficients are obtained either by angle-head centrifugation or by analytical centrifugation. In developing the mathematics of differential centrifugation in the present paper, however, it really does not make much difference whether Pickels or Anderson is right, for a logarithmic scale is used to present the data, and even a factor of two difference would shift the curves only a little and not much affect their shapes. For making the computations, however, we will adopt the approach of Anderson because of its great simplicity and also because it now appears to have better experimental support. Figure 1 shows an angle-head centrifuge tube in place, as well as the nomenclature that will be used in describing radial positions. The meniscus position is located at a, and the tube bottom is located at b. Anderson’s approach is to assume that sedimentation is strictly radial, but that material which sediments into the side walls is resuspended by Corioli forces acting to swirl the liquid in a vertical plane. Hence the position of the migrating boundary, r, is given by the simple relation T = a exp[s(&dt] (1) where s is the sedimentation coefficient of the macromolecules, o is the angular velocity, t is time, and the limits of integration cover the duration of the run.
FIG. 1. Cross-sectional drawing of centrifuge tube in angle-head rotor. Distances from axis of rotation to meniscus (a), tube bottom (b) , and any arbitrary point (T) are illustrated by arrows.
DIFFERENTIAL
CENTRIFUGATION
IN ANGLE-HEAD
DEVELOPMENT OF THE MATHEMATICS DIFFERENTIAL CENTRIFUGATION
ROTORS
281
OF
Two quantities essential for the development of the theory are q, the fraction of macromolecules remaining in solution during a centrifuge run, and p, the fraction pelleted. Since a given macromolecule must be either in the supernatent solution or else in the pellet, it follows that p + q = 1. If the sedimenting boundary has moved to a position T, then the fraction of material remaining in solution is given as :
Equation 2 is only approximately correct because the diluting effect of a centrifugal field is ignored, and because no provision is made for the hemispherical tube bottom. For our purposes, however, it is accurate enough. Values of r to be used in Equation 2 may be computed from equation 1. In order to use equation 1, s must be known. The usual conversion formula is employed to compute values of s from values of the standard sedimentation coefficient, s: (1 - VP) 0.01002 (3) 7) s = s20sw( l- ii 0.9982) where V, p, and 7, are partial specific volume of macromolecule, density of solution, and viscosity of the solution, respectively. The quantities 0.9982 and 0.01002 are the density (gm/cc) and viscosity (poise) of water at 20°C. A series of values for q, and hence p, may be computed as a function of spD,W for any particular solvent, rotor, speed and time. This is accomplished through the use of equations 3, 1, and 2, in that order. Negative values of q should be replaced by 0. Next, it is convenient to plot this series of values for p or q as a function of log szo,,,,.In Figure 2, for example, we show the results of a plot of p for a centrifuge run at 19,000 rpm, a = 4.4, and b = 8.9 cm, at 5” in a solvent having the viscosity and density of water at that temperature. The run is of 2 hr duration. A plot of p vs. .se,,,w, as drawn in Figure 2, graphically portrays that fraction of material found in the pellet at each particular sedimentation value. The construction of such a curve is desirable for the high-speed centrifuge run in which the virus is to be found in the pellet. On the other hand, for a low-speed run in which the pellet contains mostly debris and is to be discarded, while the supernatant
282
SCHUMAKER
AND REES
FIG. 2. Plot of fraction of material pelleted, p, as function of sedimentation coefficient, 820,~. Equations 3, 1, and 2 were employed, in that order, to calculate q, and hence p, for each value of ~20,~. Conditions for the run: 2 hr at 19,000 rpm, a = 4.4 cm, b = 8.9 cm, at 5°C in a solvent having the viscosity and density of water.
contains the virus and is saved, a plot of Q vs. szo,,+. gives the fraction of material at each particular sedimentation coefficient that is retained in the supernatant fluid. Let us distinguish values of p and q for high- and low-speed runs by the use of the prime (‘). Thus, let q’ be the value of q for a lowspeed centrifuge run, but retain the symbol of p as the values of p for a high-speed centrifuge run. The product of these two quantities, p$, then yields that fraction of material retained in an entire cycle composed of a high-speed run together with a low-speed run. We may immediately extend this treatment to include a number of cycles. A particularly simple expression results if the centrifuge run conditions for each of n cycles is identical. In this case, the product is simply raised to the n power: (pq’)“. It yields the fraction of material of each sedimentation coefficient recovered at the end of the experiments. A plot of this quantity is shown in Figure 3 for 1,3, and 5 cycles of differential centrifugation. The goodness or poorness of the isolation procedure for a given set of run conditions, such as rotor speed, time, etc., becomes immediately evident from an inspection of this type of plot (Fig. 3). For example, it is particularly evident that, for 3 and 5 cycles, the “high-speed cutoff” between 30 and 50 S is reasonably sharp. On the other hand, the “low-speed cut-off” extends from about 200 to 10,000 S. This result is typical of those found for alternate high- and low-speed runs. The “high-speed cut-off” increases in sharpness as the num-
DIFFERENTIAL
CENTRIFUGATION
IN ANGLE-HEAD
ROTORS
283
0 FIG. 3. Plot of fraction recovered, (pq’) n, plotted as function of SZO,Wfor n = 1 (solid line), 3 (dashed line), and 5 (dotted line) cycles of centrifugation. Conditions for the run: 120 min at 40,000 rpm at 5°C to form the pellet. The virus was then washed and clarified at 6,000 rpm for 30 min. Rotor dimensions are a = 4.8 cm, b = 8.0 cm. The solvent is assumed to have the viscosity and density of water.
ber of cycles is increased, while just the opposite situation occurs at the other end of the curve. We would like to point out, however, that by a simple change in the differential centrifugation procedure it is possible to eliminate the broad spread at the “low-speed cut-off,” and to make both flanks of the curve equally sharp. In order to do this, it is necessary to choose a much higher speed for the “low-speed” run. In this case, the initial “low-speed” pellet will contain some virus as well as debris. Therefore, it is necessary to save both the pellet and the supernatant fluid. The pellet is resuspended in about l/s of the original volume or less of fresh solvent, and again centrifuged at the same “low speed.” The supernatant fluid is saved, and the second pellet is again resuspended and centrifuged. The original supernatant fluid is now recombined with that obtained in these two washes. The mathematics of the recovery process is simple. We may write that the first “low-speed” pellet contained a fraction, p’, of any component, The second pellet, having been washed once, contained only P’~, and the third pellet, after being washed a second time, contained only P’~. Thus, when the washes are recombined with the original supernatant fluid, the amount of each component retained in the recombined supernatant fluids must be (l-~‘~). The combined supernatant fluids are then centrifuged at a high speed to
284
SCHUMAKER
AND REES
pellet the virus. This high-speed pellet is resuspended and twice again repelleted at this same high speed. The fraction of each component conserved in the final pellet from the three high-speed runs will be p3. The total recovery from both the high and low speed runs will be p3 (1 -p’“) . In Figure 4 is shown the resulting plot for the run conditions listed in the legend. Both flanks of the curve are now sharp. Therefore, this modified technique should yield greater amounts of more effectively purified material,
FIG. 4. Plot of fraction recovered, p3 (1 - p’) 3, plotted as function of SZO,~, for 3 cycles of centrifugation. A much sharper “low-speed cut-off” is obtained by this multiple washing procedure. Most virus will be isolated between 100 and 2000 Svedberg units, a region that can be obtained in good yield by appropriate selection of rotor speed, time, etc. Conditions chosen for the illustration: 13,500 rpm at 30 min for low-speed runs and 38,000 rpm at 120 min for highspeed runs, in a rotor with dimensions a = 3.8 cm and b = 8.1 cm. DISCUSSION
Quite apart from the theory presented here, the conditions chosen for the initial cycle may be selected to first provide extensive clarification of the solution to reduce the viscosity of the medium and then to concentrate the virus from an initially huge volume. After this preliminary preparation has been accomplished, however, repeated cycling is thought to yield purer and purer virus. The major advantage of repeated cycling appears to come from sharpening the flank of the “high-speed cut-off” side of the recovery plot, as shown in Figure 3. Unfortunately, this procedure will cause a broadening of the “low-speed” flank.
DIFFERENTIAL
CENTRIFUGATION
IN
ANGLE-HEAD
ROTORS
285
It is predicted by the theory presented in this communication that a much better recovery is to be expected by choosing somewhat higher “low-speed” run conditions, and then repeatedly washing the pellet which is obtained. In this manner, both a sharp “high-speed” and a sharp “low-speed” flank to the curve will be obtained, as is shown in Figure 4. If soft pellets are obtained which are readily restirred into the solution, then the best technique to employ is to withdraw the solution gently by pipet to a point considerably above the pellet. Now simply by increasing the number of cycles of centrifugation, the desired degree of purification is readily obtained, without the danger of disastrous loss of material. By suitable change in equation 3, this case can be handled mathematically. The mathematics of differential centrifugation as described in this communication are, at best, only a guide, for unexpected sedimentation behavior of macromolecules often occurs during the purification process. For example, sometimes virus will aggregate when purified of low molecular weight proteins (8). Nevertheless, it is hoped that the theory may serve useful purposes in the initial selection of optimal conditions for virus isolation, and also for correct interpretation of unusual results when they do arise. SUMMARY
Differential centrifugation is much used for the isolation and purification of virus. Appropriate conditions of rotor speed, time, number of cycles, etc. are usually chosen empirically. Simple equations are presented here for the prediction of the amount of material that will be recovered in each sedimentation class. These equations suggest choices of centrifugation conditions for improved yield and purity, REFERENCES 1.
RICHARDS,
E.
G.,
AND
SCHACHMAN,
H.
K.
(1957),
Am.
Chem. Sot. 79,
5324. 2. 3. 4.
STANLEY, W. M., AND WYCOFF, R. W. G. (1937)) Science 85, 181. SHARP, D. G. (1953)) in “Advances in Virus Research” (K. M. Smith M. A. Lauffer, eds.), Vol. I, 277. Academic Press, New York. STEERE, R. L. (1959), in “Advances in Virus Research” (K. M. Smith M. A. Lauffer, eds.) , Vol. VI, 3. Academic Press, New York. PICKELS, E. G. (1943)) J. Gen. Physiol. 26, 341. ANDERSON, N. G. (1968), Anal. Biochem. 23, 72. CHARLWOOD, P. A. (1963), Anal. Biochem. 5, 226. BRAKKE, M., AND STAPLES, R. (1958), ViTology 6, 14.
p,
p.
5.
6. 7. 8.
and and