Hi everyone. I'm trying to reanalyze the data of an experiment I did using a bayesian approach, but the complexity of the problem has me stuck. I read Richard McElreath's "Statistical Rethinking" (2nd ed.) for useful insight and I think I'm conceptually 80% of the way there, but I could use some help.
Background:
Two molecules, A and B, interact. We can probe the stability of interaction by pulling them apart until the interaction breaks, and record the rupture force (F) and the lifetime of interaction [t(F)]. The simplest models of interaction assume a single energy barrier between the bound (A-B) and unbound (A B) states. These single-barrier models are summarized by the Dudko-Hummer-Szabo (DHS) equation (picture below).
Truly a behemoth. t0 is the lifetime at zero force. x is the distance to the peak ("dagger" symbol) of the energy barrier, and delta G (just G for convenience) is the energy difference from the low energy A-B state to the peak of the barrier. All three of those must be positive. v is the shape factor (the bane of my existence). Different values of v describe different shapes (models) of the barrier. v can be either 1/2, 2/3 or 1 (maybe other values, but that is harder to interpret).
The problem:
Molecule A has a mutant, A', and we have reason to believe that the interaction A'-B is less stable than A-B, so we wish to compare parameters (t0, x and G) between the two. I've tried to turn this problem into a GLM, which is easy for v=1, but not for the other values (I think). My two main problems are that I don't know if I should treat v as a parameter of regression or treat each v as an independent model to be compared (via WAIC for example), and that I don't know how to assign a prior to v, or how to write the likelihood distribution for t(F) in the case I consider v to be a parameter (Dirichlet dist. maybe?).
Summary of variables:
*Data
-t(F)_i
-F_i
-categorical variable "mutant" (data from either A or A')
*Parameters (for A or A')
-t0_i > 0
-x_i > 0
-G_i > 0
-v_i? = 1/2, 2/3 or 1
*Relevant questions (goals of the analysis)
-Which value of v gives the best fit for A and A'? Is it the same or different for both?
-How do the physical parameters t0, x and G (if v is different from 1) compare between A and A'?
I'm thinking about using Rstan to tackle the coding, like in the book (I think Python+PyMC3 is a good, but harder to use alternative). My data so far is a little bit limited (n=6 and 9 points of t(F) vs F for A and A' respectively) so I'm not hoping for an excellent fit. I'm expecting to return to the lab soon and gather more data points. Any help is greatly appreciated. Please note that I'm not an expert on mathematical notation, but I'll do the effort to understand any and all help posted here. Thanks in advance for your time!
Background:
Two molecules, A and B, interact. We can probe the stability of interaction by pulling them apart until the interaction breaks, and record the rupture force (F) and the lifetime of interaction [t(F)]. The simplest models of interaction assume a single energy barrier between the bound (A-B) and unbound (A B) states. These single-barrier models are summarized by the Dudko-Hummer-Szabo (DHS) equation (picture below).
Truly a behemoth. t0 is the lifetime at zero force. x is the distance to the peak ("dagger" symbol) of the energy barrier, and delta G (just G for convenience) is the energy difference from the low energy A-B state to the peak of the barrier. All three of those must be positive. v is the shape factor (the bane of my existence). Different values of v describe different shapes (models) of the barrier. v can be either 1/2, 2/3 or 1 (maybe other values, but that is harder to interpret).
The problem:
Molecule A has a mutant, A', and we have reason to believe that the interaction A'-B is less stable than A-B, so we wish to compare parameters (t0, x and G) between the two. I've tried to turn this problem into a GLM, which is easy for v=1, but not for the other values (I think). My two main problems are that I don't know if I should treat v as a parameter of regression or treat each v as an independent model to be compared (via WAIC for example), and that I don't know how to assign a prior to v, or how to write the likelihood distribution for t(F) in the case I consider v to be a parameter (Dirichlet dist. maybe?).
Summary of variables:
*Data
-t(F)_i
-F_i
-categorical variable "mutant" (data from either A or A')
*Parameters (for A or A')
-t0_i > 0
-x_i > 0
-G_i > 0
-v_i? = 1/2, 2/3 or 1
*Relevant questions (goals of the analysis)
-Which value of v gives the best fit for A and A'? Is it the same or different for both?
-How do the physical parameters t0, x and G (if v is different from 1) compare between A and A'?
I'm thinking about using Rstan to tackle the coding, like in the book (I think Python+PyMC3 is a good, but harder to use alternative). My data so far is a little bit limited (n=6 and 9 points of t(F) vs F for A and A' respectively) so I'm not hoping for an excellent fit. I'm expecting to return to the lab soon and gather more data points. Any help is greatly appreciated. Please note that I'm not an expert on mathematical notation, but I'll do the effort to understand any and all help posted here. Thanks in advance for your time!
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