Phases of Diapause Exit

Introduction

Trying to make sense of Diapause exit. I’m imagining diapause exit as when organisms meet favorable conditions and then resume growth. When organisms meet favorable conditions, they first respond by terminating diapause (Kostal 2006) and and then resume growth. This trajectory can occur in a step wise fashion ( 2 phase, logistic then exponential) or continuously (1 phase, exponential). Here, I’m trying to make sense of these phases in a function-valued trait framework. The trajectory of diapause can be mapped by tracking metabolic rate, which can be measured continuously. Metabolic rate as a function of time has a certain shape then.

ref:

Kostal, V. (2006). Eco-physiological phases of insect diapause. Journal of Insect Physiology, 52, 113–127. https://doi.org/10.1016/j.jinsphys.2005.09.008

One Phase Exit

ref: Wadsworth, C. B., Woods, W. A., Hahn, D. A., & Dopman, E. B. (2013). One phase of the dormancy developmental pathway is critical for the evolution of insect seasonality. Journal of Evolutionary Biology, 26(11), 2359–2368. https://doi.org/10.1111/jeb.12227

Wadsworth model for exponential increase:

\[R = e^{(t-a)^b} + c\] where:

• a = presistent metabolic increase that defines the start of termination
• b = rate of metabolic increase during diapause termination
  
• c = initial metabolic rate during diapuse maintenance phase

Lets try to model this to see how these parameters can change the trajectory of diapause exit.

Creating the plotting function

Wad<-function(t=seq(1,60),a=12.9,b=.26,c=.95){
  #return(exp((t-a)^b)+c)
  return(exp((t-a))^b+c)
}
#Wad()

#orig function
plot(seq(1,60),Wad()*10^-4,ylim=c(0,4),xlim=c(1,90),ylab="metabolic rate")

Messing with the a parameter

#messing with a
plot(seq(1,60),Wad()*10^-4,ylim=c(0,4),xlim=c(1,90),ylab="metabolic rate")
lines(seq(1,60),Wad(a=1)*10^-4,ylim=c(0,4),col="blue",lwd=5) # shifts curve left
lines(seq(1,90),Wad(a=34,t=seq(1,90))*10^-4,ylim=c(0,4),col="red",lwd=5) # shifts curve right

Larger values of a shifts the curve over to the right, higher values of time. I’m using similar values as the paper, but not getting the same curve values. Relative differences still there.

Messing with b parameter

# b parameter
plot(seq(1,60),Wad()*10^-4,ylim=c(0,4),xlim=c(1,90),ylab="metabolic rate")
lines(seq(1,60),Wad(b=.2)*10^-4,ylim=c(0,4),col="blue",lwd=5) 
lines(seq(1,60),Wad(b=2,t=seq(1,60))*10^-4,ylim=c(0,4),col="red",lwd=5) 

Larger values make the curve steeper and shifts the curve to the left, earlier in time. Why would they include this parameter, no clue.

Messing with the c parameter

plot(seq(1,60),Wad()*10^-4,ylim=c(0,.002),ylab="metabolic rate",xlim=c(1,1.5),type="n")
lines(seq(1,60),Wad()*10^-4,col="black",lwd=5)
lines(seq(1,60),Wad(c=.1)*10^-4,ylim=c(0,4),col="blue",lwd=5) 
lines(seq(1,60),Wad(c=10,t=seq(1,60))*10^-4,ylim=c(0,4),col="red",lwd=5) 

Just adjusts the y intercept

Two Phase Exit

ref: Ragland, G. J., Fuller, J., Feder, J. L., & Hahn, D. A. (n.d.). Biphasic metabolic rate trajectory of pupal diapause termination and post-diapause development in a tephritid fly. https://doi.org/10.1016/j.jinsphys.2008.12.013

In this paper, Rhagoletis first undergo an logistic increase in metabolic rate, followed by an exponential increase.

Ragland biphase model:

\[R = \frac{\alpha}{(1+\beta e^{-t})} + c e^{at} + b\] where:

• R = respiration rate
• t = time in days out of witner treatment
• c = fitted scaling parameter
• \(\alpha\) = resp rate at the transition between logistic to exponential increase
• \(\beta\) = the timing of the initial logistic increase
• a = timing of transition between logistic and expo increase
• b = diapause baseline

where R is the respiration rate, t the time in days out of winter treatment, c is a fitted scaling parameter, and alpha, beta, a, and b are fitted parameters that determine respiration rate at the transition between the logistic and exponential increase, the timing of the initial logistic increase, the timing of the transition between logistic and exponential increase, and the diapausing baseline respiration rate, respectively.

Creating plotting function

Rag<-function(t=seq(1,100),c=.1,beta=2000000,alpha=500,a=.09,b=1){
  R= ((alpha)/(1+(beta*exp(-t))))+c*exp(a*t)+b
  return(R*10^-3)
}
#Rag()
plot(Rag(),type="n",ylim=c())
lines(seq(1,100),Rag(),lwd=3)

Logistic increase

Messing with alpha , \(\alpha\)

plot(Rag(),type="n",xlim=c(0,150),ylim=c(0,2))
lines(seq(1,100),Rag(),lwd=3)
lines(seq(1,100),Rag(alpha=250),lwd=3,col="red")
lines(seq(1,100),Rag(alpha=1000),lwd=3,col="blue")

Modulates the plateau phase or stationary phase of logistic part of curve

Messing with beta, \(\beta\)

plot(Rag(),type="n",ylim=c(0,.75),xlim=c(0,50))
lines(seq(1,100),Rag(),lwd=3)
lines(seq(1,100),Rag(beta=.1),lwd=3,col="red")
lines(seq(1,100),Rag(beta=2000),lwd=3,col="blue")

Modulates rate of increase for the first exponential part

Exponential increase

Messing with c, scaling parameter

plot(Rag(),type="n",ylim=c(),xlim=c(0,150))
lines(seq(1,100),Rag(),lwd=3)
lines(seq(1,100),Rag(c=.01),lwd=3,col="red")
lines(seq(1,100),Rag(c=.2),lwd=3,col="blue")

Modulates the exponential increase

Messing with a, timing of transition between logistic and expo increase

plot(Rag(),type="n",ylim=c(),xlim=c(0,150))
lines(seq(1,100),Rag(),lwd=3)
lines(seq(1,100),Rag(a=.01),lwd=3,col="red")
lines(seq(1,100),Rag(a=.2),lwd=3,col="blue")

Modulates the timing of exp increase

Thoughts

Exit to diapause influences the emergence patterns in wild populations and is therefore an important phenotype for understanding how organisms can differentiate throughout a season. It looks like organisms can exit diapause differently. ECB has one phase, while Rhagoletis has two phases. Although Rhagoletis is two steps, the immediate increase in metabolic rate signifies or is a hallmark of termination. And termination itself adaptively varies among strains(ECB) and races (Rhagoletis) and may facilitate and enable speciation.