FINAL REPORT
GROUP NUMBER T2
PROJECT Number and Title: Dialysis and transport of charged molecules in model and biological systems.
DATE SUBMITTED: 9th November 2002
ROLE ASSIGNMENTS
ROLE GROUP MEMBER
FACILITATOR……………………….. Sylmarie Arroyo
TIME & TASK KEEPER……………… Zaishao Cheng
SCRIBE……………………………….. Ania Oldakowska
PRESENTER…………………………. Samuel Kahn
Abstract
During week 1 of this lab calibration curves were collected for both KCl and NaCl to determine their respective linear relationships between conductance and concentration. This data was also used to determine values for NaCl and KCl, which had a +8.73% and a +3.24% deviation from the literature respectively. The linear equations were used to determine concentration in a continuously flowing dialysis cell in order to demonstrate a mass balance. KCl was run parallel to water to demonstrate simple diffusion and then parallel to Sodium polyacrylate to demonstrate the effects of Gibbs Donnan equilibrium. Data for the latter experiment was collected by running aliquots through an AA Spec. The lab culminated in comparing data collected with both the AA Spec and with conductance meters for cartilage, tendon, and meniscus plugs in order to determine the effects of GAG, a large, fixed, negative charge that is located in high concentrations in the cartilage. Our data showed that the cartilage, meniscus and tendon did not fit the theoretical sponge model. Objectives
to perform a mass balance on a dialysis system using conductance and spectroscopy to perform continuous concentration checks. to use a dialysis system to model a Gibbs-Donnan equilibrium. to characterize cultured cartilage plugs in terms of diffusion of ions in salt solutions and compare them to tendon and meniscus plugs as well as mathematical models of predicted behavior. to determine the molar conductance values for infinite dilution within 10% of the literature values.
BACKGROUND
The membrane of a cell provides a medium for the transfer of ions and other solutes to and from the cell. The driving force for the diffusion of these solutes is a difference in chemical potential, which also determines the rate of transport. A man-made membrane could be used to illustrate the arrival at equilibrium using solutions with ions to which the membrane is permeable as well as solutions with charged proteins to which the membrane is not permeable. In the second case a Gibbs-Donnan equilibrium would be expected. 1 Because of the fixed negative charge on one side of the membrane, the ions must run down their gradients to come into equilibrium. The Donnan Rule holds which states that at equilibrium (charge neutrality) the products of the diffusible ions on the two sides of the membrane are equal hence, [A][C]=[A2][C2], where A is anion, C is cation . The Gibbs-Donnan equilibrium holds true in most cells where the charge on the inside is fixed. Articular cartilage is an example of a biological tissue containing a highly-negatively charged fixed and is much easier to study than living cells, because of its size and its insusceptibility to breakage. Articular cartilage consists of cells (chondrocytes), collagen fibrils, proteoglycans (PGs), and other glycoproteins. The key control device of the PGs is the GAG chain. The negatively charged GAG serves to attract the positive counter-ions (typically Na+) found within the interstitial fluid.2 Therefore by observing the change in the concentration of Na+ ions in the surrounding fluids, a Gibbs Donnan equilibrium can be modeled. The osmotic pressure created results in swelling pressure, which can then be observed by varying the saline bath concentration. The tensile forces of the cartilage network balance this swelling pressure when no external load is applied3. This balance between the swelling pressure and the collagen network forces cause an equilibrium state under external loading4 which is a very important part of resistance to compressive forces which could otherwise cause significant damage in joints.
THEORY AND METHODS OF CALCULATION
Initially a method of determining the concentration of solution using conductances was explored. A graph of conductance vs. concentration was plotted and a relationship between the concentrations we were expecting and the conductances that we found were established. In the concentration interval that we used, it was found to be linear. The slope of this curve we defined as Λ0. We used this method of plotting conductance vs concentration curves to find the concentration corresponding to the conductance that we found in our Gibbs-Donnan experiments.
Graphs of conductance versus the square root of the concentration were done for KCl and NaCl in order to ½ obtain the equivalent conduction. The equation = - b(c) was used to check the validity of our data. By converting the measured values of conductance into the molar conductance (a graph of against (c)
1 http://entochem.tamu.edu/Teaching/Gibbs-Donnan/G-DMovieText.htm 2 http://www.health.uottawa.ca/biomech/courses/pht1507/cartilage.pdf 3 http://asme.pinetec.com/bio1999/data/pdfs/a0142080.pdf 4 http://dir2.nichd.nih.gov/nichd/stbb/0507a.pdf 1/2 was plotted. The graph was then extrapolated back to zero concentration and at infinite dilution,was determined. This method provides a way for us to compare our value of with literature
In the process of finding a model for cartilage in solution we derived the formulas
Cb+Acc-Acb’=Cb’ Cb’=(1/(1+A)Cb+(A/(1+A))Cc where in the plot of Cb’ against Cb the slope of the graph is 1/(1+A) and the intercept is A/(A+1)Cc In order to find the mass transfer coefficient k m we derived the formula :
2 0 -V 0 0 kmAx t + (C H) = -V ln (C L + C H – 2CL)
0 0 2 kmAx 0 -V Where in the plot of ln (C L + C H – 2CL) vs. t The slope is - /V and the intercept is - (C H) /V
Where Ax and V are both known constants. The full derivation of this formula can be found in the appendix.
METHODS AND PROCEDURE:
In the first week our goal was to learn to use the dialysis system provided and find a mass balance in the system. We first found the conductance data NaCl and KCl solutions at concentrations of 0.02, 0.05, 0.08,
0.5, 1 M. These conductance points were plotted against the concentrations to yield the value of Λ 0. Next, a dialysis system with distilled water on one side and 0.1 M solution of KCl on the other was constructed. Over time data of the conductance versus time was recorded for the diffusion process of the membrane until steady state was reached.
Weeks 2’s procedure consisted of dialyzing a solution of 0.1 M KCl on one side of the dialysis membrane against 1% sodium polyacrylate solution on the other side. This set up a Gibbs Donnan equilibrium. During the running of the dialysis we took samples of each solution in intervals of 15 mins. These samples were then tested in the absorbance meter to find the concentration of Na+ and K+ ions present. The conductance testing program from week 1 was ran through the duration of this experiment as well.
During week 3 we performed a dissection on a cow leg in order to obtain cartilage plugs. Seven cartilage plugs of diameter around 9 mm were extracted from the cartilage on the same bone in the cow knee joint. Two tendons and the meniscus were cut out and saved for examination in week four. Five of the plugs were placed into 20mL saline solutions with concentrations of 0.01 M, 0.05 M, 0.15M (2plugs), and 0.5 M NaCl. Over a duration of 2 hours, the wet weight of the plugs was recorded (dry and slightly dampened originally) every 15 minutes along with conductance readings. The plugs were then left in their respective solutions at room temperature.
In week 5 the five solutions containing immersed plugs used previously were put through the same conductance and absorbance testing. The plugs were all individually weighed and then disposed of. Four plugs of tendon and four plugs of meniscus were prepared in a similar manner to that of the cartilage plugs. These were immersed in saline solutions prepared using a 100 mL volumetric flask of the same concentration as the cartilage. The volume of the solution was kept constant at 20 mL. The mass of each plug was recorded every 15 minutes for duration of 2 hours as before. At each 15 minute time interval a conductance was recorded for each of the solutions. The data for tendon plugs and meniscus plugs was collected and compared to that previously found for the bone cartilage. RESULTS:
Figure 2: Potassium Gibbs Donan Effect 1.0
0.9 )
y 0.8
t i r a
l 0.7 o
m
( 0.6
n o i 0.5 t a r t 0.4 n e
c
n 0.3 o C
+ 0.2 KCl solution K Sodium Acrylate solution 0.1
0.0
0 2000 4000 6000 8000 10000 Time (s)
Figure 1: This figure shows the Gibbs Donan effect of potassium as it diffuses across a semi permeable membrane with time. Y-error bars are based on systematic and mechanical errors from volumetric equipment used in dilutions as well as variation in absorbance readings. Variation in absorbance readings was minimized by increasing integration time on the machine and by running the standard solution through the machine before every trial. A typical standard deviation from 17 trials was 0.0018, or 2.2 %. X-error bars are not shown because they are negligible. Figure 2: Equivalent Conductance vs. Square Root of Concentration of NaCl 115
110 NaCl 105 UCL and LCL
100
) (
e
c 95 n
a t c
u 90 d
n o
C 85
t n e l 80 a v i u q 75 E
70 0.0 0.2 0.4 0.6 0.8 1.0 Concentration0.5 (Molarity0.5)
Figures 2 shows the plot to determine the equivalent conductance and of NaCl. A similar plot of KCL is shown in the appendix (Figure 8). These values, and their comparison to the literature values and the implications of this, are discussed in the analysis. Figure 4: [Na+] in NaPr with KCL dialysis
Concentration of Na+ in NaPr 1.0 Concentration of Na+ in KCL
0.8 ) L
/ g
m 0.6 (
n
o i t a r
t 0.4 n
e c n o
C 0.2
0.0
0 2000 4000 6000 8000 10000 Time (s)
Figure 3 shows the concentration of Na as a function of time from the trial in which KCl was run parallel to Sodium Polyacrylate.
Final concentration Vs initial concentration for different NaCl solns with meniscus plugs
0.25
)
L 0.20 / l o
m Confidence Limits (
n Linear Fit o
i 0.15 t
a r t n e c Y=0.35802X+0.07272 n 0.10
o 2
C R =0.6889
l
a P=0.10977
n i
F 0.05
0.00 0.0 0.1 0.2 0.3 0.4 0.5 Initial Concentration (mol/L)
Figure 4 shows the plot of final concentration vs initial concentration of different NaCl solutions with meniscus plugs. A model of cartilage acting as a sponge in NaCl solution was solved and the equation relating final concentration and initial concentration was derived: Cf = Ci/(1+A) + [A/ (1+A)]Cc where 1/(1+A) is the slope and [A/(1+A)]Cc , the y intercept. (derivation of the equation is shown in the appendix). Linear fit from t=1s to t=100s -2.310 Linear fit from t=1s to t=300s Linear fit from t=1s to t=600s -2.315
-2.320 Y=-9.6987EE-8X-2.31071 Y=-1.70487E-6X-2.31109 )
L -2.325 Y=-8.51141E-8X-2.31429
C 2 - 1 . -2.330 0
( n l -2.335
-2.340
-2.345 -50000 0 50000 100000 150000 200000 250000 300000 350000 400000 time2 (sec2)
2 Figure 5 shows the plot of ln(0.1-2CL) Vs time in the determination of km. The linear fit was done using difference ranges of time to determine whether performing the experiment for a shorter period of time gives us an accurate value of km. The values of km for 0-600 sec was 2.14186E-5, for 0-300 sec was 2.356E-5, and for 0-100 sec was 4.3051E-5. Calculations are shown in the appendix. The value of km of data taken from 0-300 sec gives a +9.89% deviation compared to the value of km of the data taken from 0-600 sec (which we take as the most accurate value)and from 0-100 sec, a +101.0% deviation was obtained.
Analysis
In figure 1, the mass balance of the KCl with D.I water dialysis was obtained by converting the conductance into concentration through a calibration curve of KCl (Figure 1 in the appendix). Few data points for the calibration curve were taken because previous groups had already determined this; the linear regression from this graph was then used to convert conductance readings from unknown solutions (with c < 1 M) into concentrations. The mass balance was obtained by summing up the concentrations on both sides of the membrane, given by the green line in the graph. The total concentration was found to be constant at 0.125 mol/L throughout the system.
To determine the value of a graph of equivalence concentration was plotted against concentration0.5.(Figure 2 and 8A) This method provides a way for us to compare our value of with literature. The values of of NaCl was 115 + 2.0 Ω-1equiv-1cm2 and 145 + 2.3 Ω-1equiv-1cm2 for KCl. The values were -8.73% and +3.24% from the literature value respectively.
The mass balance in week one was solved and to calculate Km the graph of ln(C0L + 2 C0H - 2CL) vs. t was plotted (figure 5)where the slope of the linear fit was - kmAx/V and 0 -V -5, the intercept = - (C H) /V. The values of km for 0-600 sec was 2.14186×10 for 0-300 sec was 2.356×10-5, and for 0-100 sec was 4.3051×10-5.(Calculations are shown in the appendix) The value of km of data taken from 0-300 sec gives a +9.89% deviation compared to the value of km of the data taken from 0-600 sec (which we take as the most accurate value)and from 0-100 sec, a +101.0% deviation was obtained. Performing the experiment from 0-300sec would not give us a large deviation from the km value of 0-600sec.Time taken for each trial could be reduced without much sacrifice of the accuracy of km determination.
To determine if the plugs act like a sponge in NaCl solution, the final concentration of the NaCl solution is plotted against the initial concentration to determine whether the relationship is linear (defined as having p<0.05). For the meniscus plugs, the linear fit had p > 0.05, hence, by our definition, the fit is not linear. The same steps were performed for cartilage plugs and tendon plugs and the graphs are shown in the appendix. (Figure 4A and 5A). For cartilage plugs, p=0.09622 and p=0.0745 for tendon plugs, both of which are >0.05. Hence, the data does not fit the model.
Conclusions
1. The values of of NaCl and KCl were 115 + 2.0 Ω-1equiv-1cm2 and 145 + 2.3 Ω- 1equiv-1cm2 respectively. The values were + 8.73% and + 3.24% from the literature value respectively.
2. A mass balance was obtained using a dialysis cell and conductance meters to determine concentration for KCl running parallel to di-water and, in a successive experiment, parallel to Sodium Acrylate.
3. The cartilage, tendon and meniscus do not fit the theoretical sponge model.
REFERENCES:
1. “An experimental study on human articular cartilage permeability,” - F. Boschetti, C. Miotti, F. Massi, M. Colombo,V. Quaglini, G.M. Peretti, R. Pietrabissa - http://wwwlabs.stru.polimi.it/the%20staff/Colombo/f-boschetti-2-21.pdf 2,.Neuro-Bio Movie - http://www.jbjs.org/ORS_2001/pdfs/0427.pdf 3. “Chondrocyte-Extracellular Matrix Interactions during osmotic swelling,” VC Mow, DN Sun, XE Guo * , CT Hung + , WM Lai – http://asme.pinetec.com/bio1999/data/pdfs/a0142080.pdf 4. “Mechanical Properties of the Collagen Network in Human Articular Cartilage as Measured by Osmotic Stress Technique,” Peter J. Basser, Rosa Schneiderman, Ruud A. Bank, Ellen Wachtel, and Alice Maroudas - http://dir2.nichd.nih.gov/nichd/stbb/0507a.pdf 5. Gosh P, Taylor TKF: The knee joint meniscus: A fibrocartilage of some distinction. Clin Orthop Rel Res 224: 52-63, 1987. Appendix
Figure 1A: KCL Conductance Vs Concentration Curve
100 KCL UCL
LCL 80
e 60 c n
a t Y=96.26904X c
u 2
d 40 R =0.99913 n
o SD = 2.34875 C
20
0
0.0 0.2 0.4 0.6 0.8 1.0 Concentration (mol/L)
Figure 1 shows a standardization curve from week 1. This curve was obtained from taking the conductance of KCl solutions prepared for this purpose.
Figure 2A: Dependence of Conductance on Concentration 4.5 of Sodium Polyacrylate
4.0
3.5
3.0 Y=3.27805X+0.14117 2 R =1 )
C 2.5 ( SD=7.58E-16
Probe 1 e
c P<0.0001 C.L.
n 2.0 a
t Probe 2 c
u 1.5 C.L. d
n o
C 1.0
Y=2.88961X+0.12461 0.5 R2=0.99956 SD=0.03706 0.0 P<0.0001 -0.5 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 % Sodium Acrylate
Figure 2A shows the standardization curve for Na-Polyacrylate. The curve was obtained by taking different concentrations of Na-Polyacrylate solutions and running them through the conductance meter.
Figure 7A Figure 3A: Week 3 Standardization Curve: 50 Conductance/Concentration Dependence of NaCl
40 Y=83.738X-0.92522
R2=0.99878 SD=0.99705
) 30 P<0.0001 C (
e c
n a t 20 c u d
n NaCl o
C Confidence Limits 10
0
0.0 0.1 0.2 0.3 0.4 0.5 Concentration (M)
Figure 3A shows the calibration curve for NaCl done in week 3.
Final Conc Vs initial Conc of different NaCL solns with tendon plugs
0.30
0.25 ) L / l o
m
( 0.20
n Linear Fit o i t Confidence Limits a r t 0.15 n e c
n
o Y=0.43506X+0.06346
C 0.10
2 l R =0.78481 a
n i P=0.0745 F 0.05
0.00 0.0 0.1 0.2 0.3 0.4 0.5 Initial Concentration (mol/L)
Figure 4A shows the plot of final concentration vs initial concentration of different NaCl solutions with tendon plugs. Final Conc Vs initial Conc of different NaCl solns with cartilage plugs
0.020 0.018 0.016 Y=0.02157X+0.00122 2 0.014 R =0.7921 0.012 P=0.09622
) 0.010 L / l 0.008 o
m 0.006 (
c 0.004 n o
c 0.002
l a 0.000 n i F -0.002 Linear Fit
-0.004 Confidence Limits -0.006
-0.008 -0.010
-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Inital Conc (mol/L)
Figure 5A shows the plot of final concentration vs initial concentration of different NaCl solutions with cartilage plugs.
Conductance Vs time (s) of different NaCl solutions with plugs 25
20 0.01 M NaCl 0.05 M NaCl
0.15 M NaCl
e 15 0.5 M NaCl c n
a t c u d
n 10
o C
5
0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100001100012000 Time (sec)
Figure 6A shows how the conductance of 4 different concentrations of NaCl solutions with meniscus plugs vary with time. Conductance Vs time (sec) for different NaCl soln with tendon plugs 22
20
18 0.01M NaCl soln 16 0.05M NaCl soln 0.15M NaCl soln
14 0.5M NaCl soln e c n 12 a t c u
d 10 n
o
C 8
6
4
2
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100001100012000 Time (sec)
Figure 7A shows how the conductance of 4 different concentrations of NaCl solutions with tendon plugs vary with time.
Figure 3:Equivalent Conductance vs. Square Root of Concentration of KCL
160
Y=-48.69012X+141.54 150
) 2 (
R =0.94994
e Probe 1 c 140 SD=6.799 n Confidence Limits a t
c 130 Probe 2
u ) ( d
Confidence Limits n e 120 o c n C
a t t n
c 110 e u l d a n v 100 i o u C q
t
E 90 Y=-46.1954X+137.9378 n 2 e l R =1 a 80 v
i SD=2.741E-14 u q 70 E
60 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Concentration0.5 (Molarity0.5)
Figure 8A shows a plot of equivalence conductance vs concentration0.5 for KCl. Modelling cartilage as a sponge in NaCl solution
Mass in = mass out
VbCb + VcCc=VbCb’+VcCc’
Making Cb’ the subject of the equation Cb’=Cb+(Vc/Vb)Cc-(Vc/Vb)Cc’
Setting A = Vc/Vb Cb’=Cb+ACc-ACc’
At equilibrium, Cc’=Cb’
Cb+Acc-Acb’=Cb’ Cb’=(1/(1+A)Cb+(A/(1+A))Cc
Therefore, by plotting Cb’ against Cb, the slope of the graph = 1/(1+A) and the intercept is A/(A+1)Cc
Balance on Low In = mxt = kmAx(CH – CL)t Out = 0 Gen = Cons = 0 dCl Acc = V ( /dt)
dCl kmAx(CH – CL)t = V ( /dt) dCl kmAx t dt = V ( /(CH – CL)) but CH varies with CL need expression of CH in terms of CL
Total Balance Acc = Gen = Cons = 0 In = Out 0 0 0 0 In = V HC H + V LC L Out = VHCH + VLCL
0 0 0 0 V HC H + V LC L = VHCH + VLCL
0 0 0 0 CH = (V HC H + V LC L – VLCL)/ VH Assume all volume equal 0 0 CH = C L + C H - CL
dCl kmAx t dt = ½ V ( /( ½ C0L + ½ C0H – Cl)
2 0 0 ½ kmAx t + I = ½ V ln (½ C L + ½ C H – CL) 2 0 0 kmAx t + I = V ln (½ C L + ½ C H – CL)
0 but at t = 0, CL = C L
0 0 0 I = V ln (½ C L + ½ C H – C L)
0 0 0 V 0 0 V I = (½ C L + ½ C H – C L) but C L = 0 I = (½ C H)
2 0 V 0 0 kmAx t + (½ C H) = V ln (½ C L + ½ C H – CL)
0 0 2 Therefore plot ln (½ C L + ½ C H – CL) vs. t kmAx 0 V 0 slope = /V intercept = (½ C H) /V this serves to check C H and V (or if we don’t know V, we can find it to find km, since we know Ax) Mass (g) Vs Time (sec) of meniscus plugs in different NaCl solns
0.75
0.70 0.01M NaCl soln 0.05M NaCl soln 0.65 0.15M NaCl soln ) g 0.5M NaCl soln (
s 0.60 s a
M 0.55
0.50
0.45 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100001100012000 Time(sec)
Mass (g) Vs time (sec) of tendon plugs in different NaCl solns 0.98 0.96
0.94 0.92
0.90 0.88 0.01M NaCl soln 0.05M NaCl soln 0.86 0.15M NaCl soln 0.84 )
g 0.5M NaCl soln
( 0.82 s
s 0.80 a 0.78 M 0.76 0.74 0.72 0.70
0.68 0.66
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100001100012000 Time (sec)