Central dogma of molecular biology:Material balance on one specific mRNADelays in synthesisCellular compartmentalization10.37 Chemical and Biological Reaction Engineering, Spring 2007 Prof. K. Dane Wittrup Lecture 15: Gene Expression and Trafficking Dynamics This lecture covers: Approach to steady state and receptor trafficking Central dogma of molecular biology: DNA Æ mRNA Æ protein transcription translation Material balance on one specific mRNA Accumulation = synthesis – degradation moles mRNACmRNA≡ cell volume ()( )mol mRNAKr≡, transcription (function of gene dosage, inducers, etc.) time cell volume cell volumeVi≡ vessel volume ()dCmRNAVi=−KCriVVγ dtrmRNAi γr≡ first order rate constant for mRNA degredation V ≡ a function of time (cells grow, divide) i Æ can’t pull out of the derivative Do the chain rule: ddVimCCKVVRNAmRNA+=dtidtr i−γrCmRNAV idCmRNA1d V=−KCirγr mRNA−C dtmRNAVidt simplify: 1d Vi=μ (specific growth rate in exponential growth) VidtdCmRNA=−KCrγμr mRNA−C dtmRNA dilution by growth term (b/c concentration is on a per-cell volume basis) Cite as: K. Dane Wittrup, course materials for 10.37 Chemical and Biological Reaction Engineering, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].dCmRNA=−KCrr()γμ+ dtmRNA at steady-state: KCrmRNA,SS= ()γr+μ transient case, analytical solution (just integrate) KCer⎛⎞1−+()μγrtmRNA=−⎜⎟()γμr+ ⎝⎠ independent of the transcription rate constant K r t mRNAC 1rγ μ+ S.S. Figure 1. Concentration of CmRNA versus time. At long times steady state is approached. Similar rate expression for the protein: (again, per-cell volume basis, analogous constants) 10.37 Chemical and Biological Reaction Engineering, Spring 2007 Lecture 15 Prof. K. Dane Wittrup Page 2 of 4 dCp=−KCpmRNA()γμp+C dtp function of time, solved for above dCpK=−Ker()1(−+()γμrtdtpp−γμ+)C ()γμr+p steady-state: d=0, t →∞ dtKKCrppS, S= ()γrp++μγ(μ)Cite as: K. Dane Wittrup, course materials for 10.37 Chemical and Biological Reaction Engineering, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].CpS, SK=p Note: K, γ vary from protein to protein and condition Cp pmRNA,SSγp+μto condition Integrate dCp: dt⎛⎞()γμ+−ee−+()γμptrp(γ+μ)−+()γμrtCCpp=+,SS⎜⎟1 ⎜⎟γγpr−⎝⎠Usually, γprγ in E. coli ln 2∼7 minutes on average. γrfor most proteins, ln 2∼hours to days. γp also, γrμ Apply assumptions to get: KKCepr1()pp=−()()−+γμt γγrp+μ Delays in synthesis time (seconds) E. coli Yeast MammalsmRNA – 1 kb gene 10-20 30-50 30-50 Protein – 400 a.a. 20 20 60-400 10.37 Chemical and Biological Reaction Engineering, Spring 2007 Lecture 15 Prof. K. Dane Wittrup Page 3 of 4 Cite as: K. Dane Wittrup, course materials for 10.37 Chemical and Biological Reaction Engineering, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].t pC Delay is generally small compared to 1pγ μ+  Figure 2. Concentration of protein versus time. However, the delay can dramatically destabilize feedback loops. Cellular compartmentalization CC→ where CC≡ for compartment 1, and p,1 p,2 p,1 pCCp,2≡ for compartment 2 prate = KC transport p,1 cell out. cyto. prod. rec. endocytosis + kkonoff Figure 3. Diagram of protein-ligand binding on the cell surface. 10.37 Chemical and Biological Reaction Engineering, Spring 2007 Lecture 15 Prof. K. Dane Wittrup Page 4 of 4 Cite as: K. Dane Wittrup, course materials for 10.37 Chemical and Biological Reaction Engineering, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month