1. Central Dogma Theory and Kinetic Models

2. A B DNA RNA PROTEIN Overview 𝑑[𝑈] 𝑑𝑡 = ( 𝛼 1 (1+ [𝑉] β ) )-[U]
Central Dogma of Biology
DNA
RNA
PROTEIN
Kinetic Models A B
𝑑[𝑈] 𝑑𝑡 = ( 𝛼 1 (1+ [𝑉] β ) )-[U]
Computational Analysis of ODEs

3. How Modeling is used 𝑑[𝑈] 𝑑𝑡 = ( 𝛼 1 (1+ [𝑉] β ) )-[U]
Experimental Implementation
How Modeling is used
𝑉 < [𝑈]
Genetic Circuit Design
Mathematical Modeling
𝑑[𝑈] 𝑑𝑡 = ( 𝛼 1 (1+ [𝑉] β ) )-[U]
𝑑[𝑉] 𝑑𝑡 = ( 𝛼 2 (1+ [𝑈] γ ) )-[V]
[𝑉]> [𝑈]
This is an example of the engineering design cycle.

4. DNA RNA PROTEIN PROTEIN PROTEIN PROTEIN Central Dogma of Biology
ribosome
TetR
gfpmut3
lacI

5. Central Dogma of Biology
DNA
RNA
PROTEIN

6. A B Kinetic Models: Mass Action Kf KR At Equilibrium
− 𝑑[𝐴] 𝑑𝑡 = 𝐾 𝑓 [𝐴]
𝐾 𝑓 = #𝑜𝑓 𝑐𝑜𝑛𝑣𝑒𝑟𝑠𝑖𝑜𝑛𝑠 𝑜𝑓 𝐴 𝑡𝑜 𝐵 𝑆𝑒𝑐𝑜𝑛𝑑 KR
− 𝑑[𝐵] 𝑑𝑡 = 𝐾 𝑅 [𝐵]
𝐾 𝑅 = #𝑜𝑓 𝑐𝑜𝑛𝑣𝑒𝑟𝑠𝑖𝑜𝑛𝑠 𝑜𝑓 𝐵 𝑡𝑜 𝐴 𝑆𝑒𝑐𝑜𝑛𝑑
At Equilibrium
− 𝑑 𝐵 𝑑𝑡 =− 𝑑[𝐴] 𝑑𝑡
𝐾 𝑒𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚 = 𝐾 𝑓 𝐾 𝑅 = 𝐵 [𝐴]
𝐾 𝑓 𝐴 = 𝐾 𝑅 [𝐵]

7. A B Kinetic Models: Basic Example Kf KR 𝑑 𝐴 𝑑𝑡 = 𝐾 𝑅 [𝐵]- 𝐾 𝑓 [𝐴]
− 𝑑[𝐴] 𝑑𝑡 = 𝐾 𝑓 [𝐴]
− 𝑑[𝐵] 𝑑𝑡 = 𝐾 𝑅 [𝐵]
𝐾 𝑒𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚 = 𝐾 𝑓 𝐾 𝑅 = 𝐵 [𝐴]
Production
Loss
𝑑 𝐴 𝑑𝑡 𝑙𝑜𝑠𝑠 = −𝐾 𝑓 [𝐴]
𝑑 𝐵 𝑑𝑡 𝑙𝑜𝑠𝑠 = −𝐾 𝑓 [𝐴]
𝑑 𝐴 𝑑𝑡 = 𝐾 𝑅 [𝐵]- 𝐾 𝑓 [𝐴]
𝑑 𝐵 𝑑𝑡 = 𝐾 𝑓 [𝐴]- 𝐾 𝑅 [𝐵]
𝑑 𝐴 𝑑𝑡 𝑃𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛 = 𝐾 𝑅 [𝐵]
𝑑 𝐵 𝑑𝑡 𝑃𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛 = 𝐾 𝑓 [𝐴]

8. Construction of a genetic toggle switch in Escherichia coli
Review of
Construction of a genetic toggle switch in Escherichia coli
Timothy S. Gardner, Charles R. Cantor & James J. Collins

9. 1 A 01000001 . . . Z 01011010 1 1 or False True No Yes Memory Cells
A memory cell saves a 1 bit of memory.
Can be combined to represent higher order data 1 A or . . .
False
True No
Yes Z
This 0 and 1 could represent true or
Iterative Design process
Voltage 1
Voltage 1

10. DNA RNA protein Design of Toggle Switch OR OR PLs1con promoter
Repressor X
terminator OR
terminator
GFP
Rbs B
LacI
Rbs E
RBS1
PLtetO-1 promoter
Ptrc-2 promoter
 (RNAP) (1msw )
 (RNAP) (1msw )
Transcription
Transcription
RNA
mRNA-GFP/lac repressor
mRNA- TetR
ribosome
cITS is heat inducible
zGFP is cloned as a second cistron
Translation
Translation OR
protein
TetR
cIts
lacI
gfpmut3
Structures sizes are not scaled the same**

11. Kinetic Models: Genetic Toggle Switch
BIOLOGICAL DESCRIPTION
MATHEMATICAL DESCRIPTION
Transcription: DNA to RNA
Transcription: DNA to RNA
𝑔 1𝑂𝑁  𝑚𝑅𝑁𝐴 1 (rate constant = K1)
𝑔 2𝑂𝑁  𝑚𝑅𝑁𝐴 2 (rate constant = K2)
Translation: RNA to Protein
Translation: RNA to Protein
𝑚𝑅𝑁𝐴 1  U + 𝑚𝑅𝑁𝐴 1 (rate constant = K3)
𝑚𝑅𝑁𝐴 2  V + 𝑚𝑅𝑁𝐴 2 (rate constant = K4)
Macromolecular Degradation
Macromolecular Degradation
ΔTime
U  0 (rate constant = K7)
V  0 (rate constant = K8)
NOTE THAT U AND V REPRESENT COMPETING REPRESSORS, NOT GFP (well U should be equal to GFP).
𝑚𝑅𝑁𝐴 1  0 (rate constant = K9)
ΔTime
𝑚𝑅𝑁𝐴 2  0 (rate constant = K10)
Circuit Design: Gene Repression
Circuit Design: Gene Repression
γ*U + 𝑔 2𝑂𝑁 ↔ 𝑔 2𝑂𝐹𝐹 (rate constant = K5, K-5)
β*V + 𝑔 1𝑂𝑁 ↔ 𝑔 1𝑂𝐹𝐹 (rate constant = K6, K-6)

12. Kinetic Models: Genetic Toggle Switch
𝑔 1𝑂𝑁  𝑚𝑅𝑁𝐴 1 (rate constant = K1)
𝑔 2𝑂𝑁  𝑚𝑅𝑁𝐴 2 (rate constant = K2)
Transcription: DNA to RNA
MATHEMATICAL DESCRIPTION
𝑚𝑅𝑁𝐴 2  V + 𝑚𝑅𝑁𝐴 2 (rate constant = K4)
𝑚𝑅𝑁𝐴 1  U + 𝑚𝑅𝑁𝐴 1 (rate constant = K3)
Translation: RNA to Protein
U  0 (rate constant = K7)
V  0 (rate constant = K8)
𝑚𝑅𝑁𝐴 1  0 (rate constant = K9)
𝑚𝑅𝑁𝐴 2  0 (rate constant = K10)
Macromolecular Degradation
γ*U + 𝑔 2𝑂𝑁 ↔ 𝑔 2𝑂𝐹𝐹 (rate constant = K5, K-5)
β*V + 𝑔 1𝑂𝑁 ↔ 𝑔 1𝑂𝐹𝐹 (rate constant = K6, K-6)
Circuit Design: Gene Repression
NOTE THAT U AND V REPRESENT COMPETING REPRESSORS, NOT GFP (well U should be equal to GFP).

13. Modeling 𝛼 1 𝛼 2 MATHEMATICAL DESCRIPTION
𝑔 1𝑂𝑁  𝑚𝑅𝑁𝐴 1 (rate constant = K1)
𝑔 2𝑂𝑁  𝑚𝑅𝑁𝐴 2 (rate constant = K2)
Transcription: DNA to RNA
MATHEMATICAL DESCRIPTION
𝑚𝑅𝑁𝐴 2  V + 𝑚𝑅𝑁𝐴 2 (rate constant = K4)
𝑚𝑅𝑁𝐴 1  U + 𝑚𝑅𝑁𝐴 1 (rate constant = K3)
Translation: RNA to Protein
U  0 (rate constant = K7)
V  0 (rate constant = K8)
𝑚𝑅𝑁𝐴 1  0 (rate constant = K9)
𝑚𝑅𝑁𝐴 2  0 (rate constant = K10)
Macromolecular Degradation
γ*U + 𝑔 2𝑂𝑁 ↔ 𝑔 2𝑂𝐹𝐹 (rate constant = K5, K-5)
β*V + 𝑔 1𝑂𝑁 ↔ 𝑔 1𝑂𝐹𝐹 (rate constant = K6, K-6)
Circuit Design: Gene Repression
Mass-Action Kinetics:
𝑑[𝑈] 𝑑𝑡 = ( 𝑘1∗𝑘3∗ 𝑘 −6 ∗[ 𝑔 1𝑂𝐹𝐹 ] 𝑘9∗𝑘6 )*( 1 ( 𝑘1 𝑘6 + [𝑉] β ) )-k7*[U]
𝛼 1
𝑑[𝑈] 𝑑𝑡 = ( 𝛼 1 ( 𝑘1 𝑘6 + [𝑉] β ) )-k7*[U]
𝑑[𝑈] 𝑑𝑡 = ( 𝛼 1 (1+ [𝑉] β ) )-[U]
𝑑[𝑉] 𝑑𝑡 = ( 𝑘4∗𝑘2∗ 𝑘 −5 ∗[ 𝑔 2𝑂𝐹𝐹 ] 𝑘10∗𝑘5 )*( 1 ( 𝑘2 𝑘5 + [𝑈] γ ) )-k8*[V]
Green= to make it dimensionless since it is used qualitatively
First term- coop repression of constative promoters.
Second term- decay
𝛼 2
𝑑[𝑉] 𝑑𝑡 = ( 𝛼 2 ( 𝑘2 𝑘5 + [𝑈] γ ) )-k8*[V]
𝑑[𝑉] 𝑑𝑡 = ( 𝛼 2 (1+ [𝑈] γ ) )-[V]

14. Analysis Stable System Unstable System Time = 0 s Time = Δt Time = 0 s
Transitioning, now these can be used to model spects of the genetic switch, like stability..

15. Analysis Stable System Unstable System Time = 0 s Time = Δt Time = 0 s
We want something stable, unlike my memory freshman year when id cram for a test.

16. DESMOS!! 𝒚 𝟏 is Stable 𝒚 𝟐 𝒚 𝟏 are Stable 𝒚 𝟐 is Stable Analysis 𝑦 1
𝑦 2
𝑦 2
𝑦 2
𝑦=𝑙𝑜𝑔(𝛼 1 )
Β= γ=2
𝒚 𝟏 is Stable
𝒚 𝟐 is Stable
𝒚 𝟐 𝒚 𝟏 are Stable
Β= γ=2
We want something stable, unlike my memory freshman year when id cram for a test.
DESMOS!!
𝑙𝑜𝑔(𝛼 2 )=𝑥

17. Computational Analysis of ODEs
RNA
PROTEIN
gfpmut3

18. INDUCERS Repressor: Inducer: If Induced: High State Low State lacI
IPTG
aTc
Temperature
TetR
cIts

19. Computational Analysis of ODEs

20. Closing Remarks
“The work on restriction nucleases not only permits us easily to construct recombinant DNA molecules and to analyze individual genes, but also has led us into the new era of synthetic biology where not only existing genes are described and analyzed but also new gene arrangements can be constructed and evaluated.”
Wacław Szybalski, 1973