Central Dogma Theory and Kinetic Models

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

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.

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

Central Dogma of Biology DNA RNA PROTEIN

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]

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]

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

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 01000001 or . . . False True No Yes Z 01011010 This 0 and 1 could represent true or Iterative Design process Voltage 1 Voltage 1

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**

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)

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).

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]

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..

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.

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

Computational Analysis of ODEs RNA PROTEIN gfpmut3

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

Computational Analysis of ODEs

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