This paper investigates a Bayesian model and a Markov chain Monte Carlo (MCMC) algorithm for gene factor analysis. Each sample in the dataset is decomposed as a linear combination of characteristic gene signatures (also referred to as factors) following a linear mixing model. To enforce the sparsity of the relative contribution (called factor score) of each gene signature to a specific sample, constrained Bernoulli-Gaussian distributions are elected as prior distributions for these factor scores. This distribution allows one to ensure non-negativity and full-additivity constraints for the scores that are interpreted as concentrations. The complexity of the resulting Bayesian estimators is alleviated by using a Gibbs sampler which generates samples distributed according to the posterior distribution of interest. These samples are then used to approximate the standard maximum a posteriori (MAP) or minimum mean square error (MMSE) estimators. The accuracy of the proposed Bayesian method is illustrated by simulations conducted on synthetic and real data.