Comparing the distributions of two processes, one of which is constrained by zero I have two continuous stochastic Markov processes: the concentration readout of two proteins in a cell over time. These are shown in this figure, where the blue line is the unbounded protein, and all other lines are zero bounded proteins, but I am interested in just one of them. (The y axis is concentration, and sorry for the poor quality image).

In an oversimplified model, the two proteins are produced under the same regime and therefore their concentration distributions (over the whole simulation period) should be identical. However the concentration of protein 1 is close to zero, so is zero-inflated.
Maybe there is a proper terminology which I am neglecting because I don't know the field, but I want to say that the distributions of the driving processes for the two proteins are the same, but the resultant protein concentration distributions are different, (Gaussian and Gamma maybe?) seen in the figure below. This is the case for my simplistic model containing these two proteins.

Now in the real biological system, experiments show a similar pattern, the concentration distribution over the observation period looks Gaussian for the non-bounded protein, but long-tailed for the zero-bounded protein. I want to know whether they are driven by the same process. Is there some way I can compare their distributions, accounting for the fact that one of them is bounded at zero?
I.e. I want to ask, is the difference in the two distributions explained fully by the fact that one of them is close to zero, or is there another difference.
Is this possible? Please let me know if I have not phrased this clearly, or need to include more information.
 A: It looks like you are using Matlab based on the graph, but I don't have access to that at the moment so the below example is in R.  I tried to generate a mixture model of a gamma and normal which looks reasonably like your data.  My thought was that after fitting this model, you could either compare the parameter estimates of the normal component to the other distribution, or run a GOF test comparing the normal component to the other data.    
Specify parameters for normal/ gamma mixture
mu <- 1000
s <- 300
p <- 0.6
shape <- 1.2
rate <- 1/ 300

Generate normal gamma mixture
whichMix <- rbinom(1000, 1, p)
mixSample <- rep(NA, 1000)
mixSample[which(whichMix == 1)] <- rgamma(sum(whichMix), shape, rate)
whichMix <- -1 * (whichMix - 1)
mixSample[which(whichMix == 1)] <- rnorm(sum(whichMix), mu, s)

Estimate mixture model
require(flexmix)
fit <- flexmix(mixSample ~ x, k = 2, model = list(FLXMRglm(mixSample ~ 1),
                                                  FLXMRglm(mixSample ~ 1, family = "Gamma")))

summary(fit)
parameters(fit)

We get estimated values reasonably close to the true mixture:
> parameters(fit)
[[1]]
                   Comp.1   Comp.2
coef.(Intercept) 226.3455 992.6114
sigma            154.6702 293.0577

[[2]]
                      Comp.1       Comp.2
coef.(Intercept) 0.004418025  0.001007444
shape            1.414647496 11.463064235

GOF test:
ks.test(mixSample[fit@cluster == 2], rnorm(1000, 1000, 300))
D = 0.058, p-value = 0.1994
alternative hypothesis: two-sided

