March 30

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