March 30

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Contents

Chemical counter

  • System
    • Input : what we want to count
      • Variable I that is continuous with respect to time, given by I=f(t)
      • Examples: iptg, light, chemical species
      • Count peak of I up to some time, T
    • Peak : I(t)'=0, I(t)>0 (maximum), above threshold, and minimum between peaks lower than another threshold
    • Chemical species : Xi
      • Either different molecule (repressors one and two)
      • Same molecule different conformation (different conformation of DNA)
      • Same molecule in a different location
    • Reaction routes
      • Some reactions are a function of I (flipping a bit)
      • Rule : bit flips when reaction takes place
    • State
      • Given by the array of species
      • Can't measure every concentration
      • Interrogate a subset of the system
    • Predict
      • N(t)
      • X(t) represents the state of X at time T, which could be Xi, Xii, etc.
      • This is a good counter : P(X(t))=X(5)=1, if t=5
    • Example
      • X0 at time=0
      • Assume discrete input with T(up) and T(low)
      • Rules:
        • Do nothing when signal is low
        • When high: xi->x(i+1) with rate r and time Tci
        • Time given by Tci = time of reaction = exponential distributed with respect to rate = Iu
        • Probability that Tci > T; Iu*r^(-Iu*t)
        • Probability that Tci > t is given by e^(-Iu*t)
        • The chance does not decrease with respect to time
    • In the time interval, Tu, there is a probability of changing from Xo to Xi
      • P(Tc>t) = e^(-Iu*t)
      • Requirment : Tu>Tc
    • What can go wrong:
      • Remain at Xo, thus Tc<Tu, with Tc driven by reaction rate given by PDF: P(tc > t)=e^(-Iu*t)
      • Key points
        • Iu needs be fast enough to allow Tc<Tu
        • Not too fast such that Tci+Tc2<Tu
    • P(correct) at first count; Tci<Tu and Tci+Tcii>Tu
      • Tci=T*, want T*<Tu
      • Tcii+T*>Tu; Tcii>Tu-T*
      • Integral ( P(Tci=T*) P(Tcs>Tu-T*) dT
        • P(Tci=T*) from PDF = Iu*-IuT*
        • P(Tcs>Tu-T*) from PDF = Iu*-Iu(Tu-T*)
    • Tl = nothing happens and Tu=pulse; only change state during Tu and Tl can be long, short, unequal, etc
    • xi->x(i+1) with rate proportional to Iu given by PDF: Iu*e^-(Iu*t)
    • P(X(k*Tu))=Xk=Probability of correct counting at count value K
  • Key rules
    • tci+tcii ...tck < kTu ; exactly k events over kTu
    • tci+ ...tc(k+1) > kTu
    • Possion process: probability of k events occur over timescale T = e^(-lamba*T)*(lambda*T)^k.(k!)
      • Interval between each event
      • Lambda is the rate at which each event occurs: Iu, same as Iu in tu ~ Iu*e^(-Iu*t)
      • Time is k*Tu
    • Probability that one counts correctly: how will this change if I change Iu, Tu,
  • Input signal changes over time
    • It peaks up at some point
    • During time interval, Tu, there is probability of state change
  • Probability of change state tci ~ Iu*e^(-Iu*t)
    • Driven species : Iu
    • More molecules
  • Successful count given by
    • tci < Tu
    • tcii+tcII>Tu
  • I want to have K state changes over total induction time K*Tu
    • Implies that intermediate state changes don't matter
      • For example two count within one large induction time and no change within the second
    • Probability of this happening is given by Poisson distribution
      • P(k, Iu) = e^(-lamba*T)*(lambda*T)^k.(k!)


Iu is the reaction rate for flipping driven by:


    • Want k spots within kTu

What data can we get now?

  • We can induce and get uni-directional flip
  • We can measure growth in fluorescent signal

What are the critical issues

  • How fast does the flippee flip?
    • Current data lumps all events, so we need to dis-aggregate

How do we de-couple the events?

  • Big question: what is the rate limiting factor in the process?
  • When is time between induction and binding?
    • Localization assay with flipper-GFP fusion
  • When is time between induction and flipping?
    • PCR with respect to time
      • Bulk assay, but don't know distribution
      • How do we do this quantitatively?
      • Now, we amplify spontaneous flipping ...
  • How fast does it take to visualize GFP?
    • Measure signal with respect to time following induction under control of same promoter


  • What drives the time of flipping?
    • Inducer length
    • Int / Xis expression dynamics

Modeling

  • Flippee
  • System

General experimental tools

  • Reporter
    • Gemini

Measure Int / Xis leakiness

  • Put reporter behind the Int or Xis promoter
    • Measure signal when not induced
  • Tag the Int with fusion
    • Direct measurement of the flipper

mRNA quantification

  • mRNA coding for flipper (int)
    • Tag mRNA with florescence to quantify

Measurement of DNA binding

  • FP-Int fusion
    • Low copy Int expression and visualize signal only when bound

Recombination

  • Bulk measurements for timescale of flipping
    • Plate reader

**PCR with respect to time

      • Induce
      • Stop reaction
      • PCR
    • How sensitive is the PCR single with respect to the number of flipped templates?
  • Single cell measurements for timescale of flipping
    • See variability across the population
    • Get the distribution
  • FP on plasmid measurement
    • Distance between two signal cases is ~100-80nm on plasmid
    • Plasmid replicates and is not synchronized with the cell
      • Therefore two dots could come from
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