1. Chapter 10. Mass Transport and Biochemical Interactions
BMOLE Biomolecular Engineering Engineering in the Life Sciences Era
BMOLE – Transport
Chapter 10. Mass Transport and Biochemical Interactions
Text Book: Transport Phenomena in Biological Systems
Authors: Truskey, Yuan, Katz
Focus on what is presented in class and problems…
Dr. Corey J. Bishop Assistant Professor of Biomedical Engineering Principal Investigator of the Pharmacoengineering Laboratory: pharmacoengineering.com Dwight Look College of Engineering Texas A&M University Emerging Technologies Building Room 5016 College Station, TX
Francis Crick and neuroscience? Why is E = mc2 and not ½ mc^2… Name all of the energy equations you know? Integral => 1/2
© Prof. Anthony Guiseppi-Elie; T: F:

2. Reaction Rates N in moles 𝑹 𝒊 = 1 𝑽 𝒅 𝑵 𝒊 𝒅𝒕 = 𝒅 𝑪 𝒊 𝒅𝒕
Ri is the reaction rate per unit volume
Moles/unit volume/unit time
𝑹 𝒊 = 1 𝑽 𝒅 𝑵 𝒊 𝒅𝒕 = 𝒅 𝑪 𝒊 𝒅𝒕
𝑹 𝒊 𝒔 = 1 𝑺 𝒅 𝑵 𝒊 𝒅𝒕
Ris is the reaction rate per unit surface area
Moles/unit area/unit time
𝒗 1 𝑨+ 𝒗 2 𝑩⇌𝑪
Reaction stoichiometry
𝑹 𝒄 = −1/ 𝑣 1 𝑹 𝑨 =−1/ 𝑣 2 𝑹 𝑩
Q = volumetric flow rate
V = volume of reactor
V/Q is = residence time
− 𝑹 𝒊 = 𝑸 𝑽 𝑪 𝒊𝒐 − 𝑪 𝒊
Not often used with proteins, peptides, and nucleotides because of the limited supply (not steady flow).

3. Reaction mechanisms 𝑹=𝒌 𝑪 𝑨 𝒂 𝑪 𝑩 𝒃 𝑪 𝑪 𝒄
𝑹=𝒌 𝑪 𝑨 𝒂 𝑪 𝑩 𝒃 𝑪 𝑪 𝒄
Generally, Rates = constant * reactants * products * catalysts and a, b, c represent
Reaction orders of individual reactants or products or catalysts and a +b +c = overall reaction order
Is k actually a constant?

4. Reaction mechanisms 𝑹=𝒌 𝑪 𝑨 𝒂 𝑪 𝑩 𝒃 𝑪 𝑪 𝒄
𝑹=𝒌 𝑪 𝑨 𝒂 𝑪 𝑩 𝒃 𝑪 𝑪 𝒄
Generally, Rates = constant * reactants * products * catalysts and a, b, c represent
Reaction orders of individual reactants or products or catalysts and a +b +c = overall reaction order
Is k actually a constant?
What affects k the most?

5. Reaction mechanisms 𝑹=𝒌 𝑪 𝑨 𝒂 𝑪 𝑩 𝒃 𝑪 𝑪 𝒄
𝑹=𝒌 𝑪 𝑨 𝒂 𝑪 𝑩 𝒃 𝑪 𝑪 𝒄
Generally, Rates = constant * reactants * products * catalysts and a, b, c represent
Reaction orders of individual reactants or products or catalysts and a +b +c = overall reaction order
Is k actually a constant?
What affects k the most?
Arrhenius equation
𝒌=𝑨 𝒆 − 𝑬 𝒂 𝑲 𝑩 𝑻
Ea = energy of activation (Joules)
A = a constant for each chemical reaction defining the frequency of collisions in correct orientation
Units of exponent?
Units of k?

6. Reaction mechanisms
Generally, Rates = constant * reactants * products * catalysts and a, b, c represent
If disappearing or appearing dictates the sign
Reaction orders of individual reactants or products or catalysts and a +b +c = overall reaction order
𝑹=𝒔𝒊𝒈𝒏 ∗𝒌 𝑪 𝑨 𝒂 𝑪 𝑩 𝒃 𝑪 𝑪 𝒄
Is k actually a constant?
What affects k the most?
Arrhenius equation
𝒌=𝑨 𝒆 − 𝑬 𝒂 𝑲 𝑩 𝑻
Ea = energy of activation (Joules)
A = a constant for each chemical reaction defining the frequency of collisions in correct orientation
Units of exponent?
Units of k?
How do you know the units of k?

7. Example:

8. Example:
Solve for Feq 1

9. Example:
Solve for Feq 2 1

10. Example: 2 1 3 solve for k Solve for Feq

11. Example:
Solve for Feq 2 1 3

14. Laplace Transformations

17. Solving differential equations via laplace
Convert differential equation to F(s) using Laplace table
Solve for F(s)
Apply initial conditions of f(t=0) and f’(t=0), etc.
Calculate any unknowns by plugging in fake s values (i.e., 0, 1, 2) to solve
Take the inverse using the Laplace table
Done
Example: y’’-10y’+9y=5t
Sometimes easier to think about using f(t) and F(s)…

18. 𝒇 ′′ 𝒕 −𝟏𝟎 𝒇 ′ 𝒕 +𝟗𝒇 𝒕 =𝟓𝒕;𝒇 𝟎 =−𝟏 𝒂𝒏𝒅 𝒇 ′ 𝟎 =𝟐
𝒇 ′′ 𝒕 −𝟏𝟎 𝒇 ′ 𝒕 +𝟗𝒇 𝒕 =𝟓𝒕;𝒇 𝟎 =−𝟏 𝒂𝒏𝒅 𝒇 ′ 𝟎 =𝟐
𝑳.𝑻. :𝓛 𝒇 ′′ 𝒕 = 𝒔 𝟐 𝑭 𝒔 −𝒔𝒇 𝟎 −𝒇′(𝟎)
𝑳.𝑻. :𝓛 𝒇′ 𝒕 =𝒔𝑭 𝒔 −𝒇 𝟎
𝑳.𝑻. :𝓛 𝒇 𝒕 =𝑭 𝒔
𝑳.𝑻. :𝓛 𝒕 = 𝟏! 𝒔 𝟏+𝟏 ; note; 0!=1 and 1!=1
𝒔 𝟐 𝑭 𝒔 −𝒔𝒇 𝟎 − 𝒇 ′ 𝟎 −𝟏𝟎 𝒔𝑭 𝒔 −𝒇 𝟎 +𝟗𝑭 𝒔 = 𝟓 𝒔 𝟐
𝒔 𝟐 𝑭 𝒔 +𝒔−𝟐− 𝟏𝟎𝒔𝑭 𝒔 +𝟏𝟎 +𝟗𝑭 𝒔 = 𝟓 𝒔 𝟐
𝒔 𝟐 𝑭 𝒔 +𝒔−𝟏𝟐−𝟏𝟎𝒔𝑭 𝒔 +𝟗𝑭 𝒔 = 𝟓 𝒔 𝟐
(𝒔 𝟐 −𝟏𝟎𝒔+𝟗)𝑭 𝒔 +𝒔−𝟏𝟐= 𝟓 𝒔 𝟐
𝒔−𝟏 𝒔−𝟗 𝑭 𝒔 += 𝟓 𝒔 𝟐 − 𝒔−𝟏𝟐
𝑭 𝒔 += 𝟓 𝒔 𝟐 − 𝒔−𝟏𝟐 𝒔−𝟏 𝒔−𝟗 = 𝟓− 𝒔 𝟑 −𝟏𝟐 𝒔 𝟐 𝒔 𝟐 𝒔−𝟏 𝒔−𝟗 = 𝑨 𝒔 + 𝑩 𝒔 𝟐 + 𝑪 𝒔−𝟏 + 𝑫 𝒔−𝟗 ;𝒔= −𝟏,𝟎,𝟏,𝟐 ;𝟒 𝒆𝒒., 𝟒 𝒖𝒏𝒌.
𝑭 𝒔 = 𝟓𝟎 𝟖𝟏 𝒔 + 𝟓 𝟗 𝒔 𝟐 + 𝟑𝟏 𝟖𝟏 𝒔−𝟗 + −𝟐 𝒔−𝟏
𝓛 −𝟏 𝑭 𝒔 =𝒇 𝒕 = 𝟓𝟎 𝟖𝟏 + 𝟓 𝟗 𝒕+ 𝟑𝟏 𝟖𝟏 𝒆 𝟗𝒕 −𝟐 𝒆 𝒕

21. Use: kcell, kne, kni, kbd equal to 7.5e-4, 0.72, 1.1, and 1.8 hr-1
P = plasmid # (note that volume is not taken into account); simply # of plasmids in a given compartment at any given time t.
Use: kcell, kne, kni, kbd equal to 7.5e-4, 0.72, 1.1, and 1.8 hr-1
kdeg1 and kdeg2 equal to 0.62
And hr-1
Where A+B at t=2 hours is total plasmid #
in cell. i.e., plasmids
Why 2 degradation rates of plasmids (polyplex context)?

22. Use: kcell, kne, kni, kbd equal to 7.5e-4, 0.72, 1.1, and 1.8 hr-1
P = plasmid # (note that volume is not taken into account); simply # of plasmids in a given compartment at any given time t.
Use: kcell, kne, kni, kbd equal to 7.5e-4, 0.72, 1.1, and 1.8 hr-1
kdeg1 and kdeg2 equal to 0.62
And hr-1
Where A+B at t=2 hours is total plasmid #
in cell. i.e., plasmids
Why 2 degradation rates of plasmids (polyplex context)?
What is the rate limiting step? Why is it what it is and not perhaps the expected
nuclear uptake step?
Why a Heaviside step function (HSSF)?
Be able to
do this

23. Use: kcell, kne, kni, kbd equal to 7.5e-4, 0.72, 1.1, and 1.8 hr-1
P = plasmid # (note that volume is not taken into account); simply # of plasmids in a given compartment at any given time t.
Use: kcell, kne, kni, kbd equal to 7.5e-4, 0.72, 1.1, and 1.8 hr-1
kdeg1 and kdeg2 equal to 0.62
And hr-1
Where A+B at t=2 hours is total plasmid #
in cell. i.e., plasmids
Why 2 degradation rates of plasmids (polyplex context)?
What is the rate limiting step? Why is it what it is and not perhaps the expected
nuclear uptake step?
Why a Heaviside step function (HSSF)?
Be able to
do this
Be able to
do this
Apply Laplace to solve…
How would you do this?
How would you deal with the HSSF without
Taking the laplace of the HSSF.

25. Why is Laplace with an e-st. : Could you use cos(st) or sin(st)
Why is Laplace with an e-st?: Could you use cos(st) or sin(st)? or any other ex form or another answer of a differential equation?

26. Why is Laplace with an e-st. : Could you use cos(st) or sin(st)
Why is Laplace with an e-st?: Could you use cos(st) or sin(st)? or any other ex form or another answer of a differential equation?
The laplace table is simplified to the greatest extent using e-st. The most basic and most common answer of differential equations is e-xt.
How do you solve more complicated partial differential equations?

27. Why is Laplace with an e-st. : Could you use cos(st) or sin(st)
Why is Laplace with an e-st?: Could you use cos(st) or sin(st)? or any other ex form or another answer of a differential equation?
The laplace table is simplified to the greatest extent using e-st. The most basic and most common answer of differential equations is e-xt.
How do you solve more complicated partial differential equations?
Assume P(x,y) = f1(x)*f2(y) => similar principle…
Laplace(s)=definite integral(f1(st)*f2(t)) and t goes away…
23 March 1749 – 5 March 1827
Kind compliments of Wikipedia.org

28. Matrix Math for Chemical Reactions
Chapter 2
Multi-Step Reactions: The methods for Analytical Solving the Direct Problem
The following slides are based on this chapter cited below…

29. Short-hand

30. Another example:
Can you do this? Try it…

31. Another example:

32. Another example: ?

33. Another example:
Now what?

34. Another example:
Multiple by dt
Separate variables properly
integrate

35. Another example: Note: k matrix is Always an nxn matrix Multiple by dt
Separate variables properly
integrate
Is this the correct notation?
Eigenvalues and vectors are for this…
Alternative?

36. More details… Remember? Same slide as presented in an earlier chapter…
For low Reynold‘s #s, the drag force is (K is translation tensor; v = velocity):
𝑭 𝑫 =𝑲 ∗𝒗
K is a symmetric tensor and the components thereof are friction coefficients fij
The principal friction coefficients are where within f?
How do we compute it? Det(K – fI) = 0 (f1 =f2 =f3 etc for an isotropic body)
Where does this come from… Av=λv…
Eigenvectors are scalable and translatable… v(A- λ?)=0… v(A- λI)=0
These are the eigenvalues and are the friction coefficients. The only non-zero solutions are
Calculated using the determinant…
For an nxn matrix, the eigenvectors will be a what? nx1…
I is a what? nxn identity matrix.
fbar = average = the harmonic mean:
1/fbar = 1/3(1/f1 + 1/f2 + 1/f3 +…)
Note that for a sphere f = 6ᴫμR… sound familiar?
S.E.

37. v(A- λ?)=0… not mathematically possible
v(A- λI)=0
Solve for λ and then using Av= λv solve for eigenvectors
Remember they are scalable and translatable and usually they are normalized to have a magnitude of 1.

38. LHS= RHS=nx1 vector (rxc) Calculate eigenvalues and vectors of k:
k = nxn matrix
X<i> = nx1 vector
λi = scalar
LHS= RHS=nx1 vector (rxc)
Calculate eigenvalues and vectors of k:
X_barbar = matrix of eigenvectors (as columns)
Multiple by dt
Separate variables properly
integrate
Matrix diagonalization…

39. Matrix diagonalization
A = PDP-1
P = eigenvector matrix
D = diagonalization of
Eigenvalues via Identity Matrix
i.e.,
k= xΛx-1 So
Capital_Lambda_barbar = lambda*Identity matrix

40. A = randn(3,3);
[V,D]=eig(A); V = 3x3 and D = a diagonal 3x3 with eigenvalues on diag.
A*V = V*D and A*V(:,1)=V(:,1)*D(1,1)
If A = symmetric then there are 3 eigenvalues (not necessarily distinct) and 3 orthogonal eigenvectors…
How to from A calculate eigenvalues and eigenvectors.
To do manually:
Ex: from A= [3 0 0; 0 3 0; 0 0 3] or any A… making it simple for example…
A-lambda*I = 0; solve for lambdas; each root is a lambda i.e., (λ-3)*(λ-3)*(λ-3)=0; all 3 are 0 but of course don’t have to be…
Then solve for V1 by A*V(:,1)=V(:,1)*D(1,1)
Then solve for V2 by A*V(:,2)=V(:,2)*D(2,2)
Then solve for V3 by A*V(:,3)=V(:,3)*D(3,3);

41. [V,D]=eig(A); V = 3x3 and D = a diagonal 3x3 with eigenvalues on diag.
A = randn(3,3);
[V,D]=eig(A); V = 3x3 and D = a diagonal 3x3 with eigenvalues on diag.
A*V = V*D; but isn’t Av=(λ=D)V?... Why is it wrong in matlab? and A*V(:,1)= D(1,1) *V(:,1)
If A = symmetric then there are 3 eigenvalues (not necessarily distinct) and 3 orthogonal eigenvectors…
How to from A calculate eigenvalues and eigenvectors.
To do manually:
Ex: from A= [3 0 0; 0 3 0; 0 0 3] or any A… making it simple for example…
A-lambda*I = 0; solve for lambdas; each root is a lambda i.e., (λ-3)*(λ-3)*(λ-3)=0; all 3 are 0 but of course don’t have to be…
Then solve for V1 by A*V(:,1)=V(:,1)*D(1,1)
Then solve for V2 by A*V(:,2)=V(:,2)*D(2,2)
Then solve for V3 by A*V(:,3)=V(:,3)*D(3,3);
Now let’s convince ourselves:
Why?

42. Is this equal to exp(V. D. m. inv(D))Co in matlab
Is this equal to exp(V*D*m*inv(D))Co in matlab? Is this equal to Coexp(Kt)? Matlab won’t let you use symbolic math with t; reserved for other things… MatLab has a hard time simplifying: V*exp(D*t)*inv(V)*Co

43. Clc k*C = dC/dt syms k1, k2, k3, k4 k*dt = 1/C dC
exp(kt)+constant = C(t)
exp(kt)*constant=C(t)
syms k1, k2, k3, k4
How do you solve for V and D manually from K?

44. Given: K = [1 -1; 2 4]; what is D and V
Caclulate compartments CA(t) and CB(t) using:
You should know though that to make sense there should be rate constants (and/or numbers) in the matrix K but for
Simplicity to show the math process K is a function of numbers only here
Be able to get to this end stage from knowing a compartmental scheme… i.e.,
I want you to be able to know every step... To get to C(t) using matrix math using chemical/rate reactions.

45. Earlier…
How?

46. A = 3x3
V = 3x3
D = 3x3
A*V = V*D and A*V(:,1)=V(:,1)*D(1,1)=D(1,1)*V(:,1) but A*V does not equal D*V

47. Principal Component Analysis
Reduces complex data => linearly uncorrelated orthogonal components
Typically: Av = λv

48. Principal Component Analysis
Reduces complex data => linearly uncorrelated orthogonal components
Typically: Av = λv
(A – 11TA(1/n) = C)TC = M
M = variance-covariance matrix
Mv = λv; v = eigenvectors; λ = eigenvalues
Mv = scores: inter-variable correlations
v = coefficients rank variables’ contributions
A = randn(3)
B=cov(A)
Stdev = sqrt(variance)
Sqrt(B(1,1)=stdev(A(:,1))

49. PCA 1) rows of samples with all the variables as columns
2) cannot be missing any data
3) normalize all of the data
4) [coefs, scores, variances, X] = princomp(normalized_data);
5) Main plots
Percentage of variance contribution
Scores plot
Loading plot
Further analysis for trends to predict

50. Normalized Variables Physico-chemical properties: Cellular function
Mn, Mw (MW*), PDI, DP, B:S, B, S, B+S
Joback/Crippen’s Fragmentation method: BP*, MP*, CV*, GFE*, LogP*, MR*, HtF*, tPSA
* = repeating unit (vs full polymer)
Cellular function
Viability, uptake, transfection

51. percent_exp = 100*variances/sum(variances)

52. Figure S1 plot(coefs(:,1:2))
Mention some variables line up with PC1 and other variables line up with PC2 and some are in between.
plot(coefs(:,1:2))
Figure 1. Top: the variances are explained for the first5 PCs. The top 4 variables of the 27 are ranked by the degree to which they contribute to the PCs
according to their associated coefficients and listed above the variances. The “(-)” indicates a negative contribution; Bottom: The loading plot of all of the
variables showing relative correlations.

53. Variables driving function
Ranking of variables: Acos(Θ) = correlation strength
E number index was most neutral in all cases: 0.02, 0.01,

54. Scores Plot
plot(scores(:,1),scores(:,2)) Mn Mw
“Self-assembled”
B+S

55. Red = highest transfection level
“Self-assembled”
The scores plot versus transfection efficacy with red being the highest level of transfection; B: region B of 3A with a B+S (sum of carbons in backbone and sidechain) value of 7;
C: region C of 3A with a B+S value of 8; D: region D of 3A with a B+S value of 9.

57. I developed an algorithm to assess whether the aortic valve is opening in rotary VADs which can be implemented in the software for RPM auto-regulation RPMs are set to a value based on echocardiography infrequently Problems with not regulating RPMs: 1) LV collapse events which inhibits blood flow 2) Calcification of aortic valve if no blood traverses normal path (aortic valve stenosis 3) Stenotic valve can result in thromboembolism and subsequent cerebrovascular accident/stroke Software would inhibition of thrombosis formation and aortic valve calcification (bridge to recovery)

58. I developed an algorithm to assess whether the aortic valve is opening in rotary VADs which can be implemented in the software for RPM auto-regulation
Fouriér Transformation
Beginnings of PCA
Resembles Arterial Pressure
Line with Dichrotic Notch

59. I developed an algorithm to assess whether the aortic valve is opening in rotary VADs which can be implemented in the software for RPM auto-regulation

60. Principal Component Analysis Dot product of PC1 and electrical signal in time
Confirmed with echocardiography

61. PCA: Predicting seizures from electroencephalograms Predicting cardiac arrhythmias from pacemakers It can be used as a learning algorithm You can use facial recognition with it You can breakdown someone’s voice and look at the strongest frequencies contributing to it and do voice recognition

62. More PK/PD (sometimes written PKPD)
ADME

65. Common variables for PK/PD
Cp=plasma concentration of drug
DB, D0=drug amount in body, initial drug concentration
Cl, ClR, ClH, ClT , ClNR=clearance, renal clearance, hepatic, total body, non-renal
VD= volume distribution of drug
R = rate of infusion into systemic circulation
k, km, ke, ka=degradation rate, metabolism rate, excretion rate, absorption rate
AUC = area under the curve (Cp(t))
F= bioavailability
Du= drug amount in urine
Ab = absorption