Enforcing non-negative gene expression dynamics

[anshu957]

Objective: Predict gene expression dynamics over time in the gene space by integrating the vector field.

Current approach: Learn vector field in PCA space, integrate the trajectory forward in time in PCA space and then project the gene expression (for all time points) back to gene expression space.

Issue: Is there a way to enforce/constraint the learned vector field function so that the solution is non-negative? I know PCA space is an abstract space so probably it is not possible there. But integrating forward with some genes going negative may result in inaccurate predictions.

Solution explored: Learn the vector field in gene space. However, learning in the gene space (let's say with ~2k genes) does not result in a good vector field re-construction despite trying multiple values of M (number of basis function). Even if the re-construction is good, how do we enforce positveness of the solution?

Any recommendation or suggestion?

[Xiaojieqiu]

@anshu957 this is an excellent question and actually can be one part of a research problem. To solve it, we may need to design some new ways for vector field reconstruction. A naive approach to solve this can be something like the following (I am using the ode_bifur2gene as an example):

from scipy.integrate import odeint

velocity_func = lambda y, t: (y>0) * dyn.sim.ode_bifur2genes(y, a=[1, 1], b=[1, 1], S=[0.5, 0.5], K=[0.5, 0.5], m=[4, 4], n=[4, 4], gamma=[1, 1])

X = []
for t in t_list:
    true_x0 = numpy.random.uniform(low=0.0, high=4.0, size=(num_of_samples,2))
    X_tmp = np.array([ odeint(velocity_func, true_x0[i], np.array([0,t]) ) for i in range(num_of_samples) ])[:,-1]
    X.append(X_tmp)

The key above is to use (y > 0) * which will make sure the integrated values always positive