We study "chromatin gels", model systems for chromatin, to theoretically predict the conditions, under which such gels show negative Poisson's ratios. A chromatin gel shows phase separation due to an instability arising from the disassembly of nucleosomes by RNA polymerases during transcription. We predict a negative Poisson's ratio near a miscibility threshold due to the cooperative assembly and disassembly of nucleosomes. The Poisson's ratio becomes more negative with an increasing number of RNAP because the disassembly rate of nucleosomes increases. In contrast, the chromatin gel shows a positive Poisson's ratio far from the miscibility threshold because the assembly of nucleosomes is arrested by the expiration of freely diffusing histone proteins.