In prokaryotes, thermodynamic models of gene regulation provide a highly quantitative mapping 3 from promoter sequences to gene expression levels that is compatible with in vivo and in vitro bio-4 physical measurements. Such concordance has not been achieved for models of enhancer function 5 in eukaryotes. In equilibrium models, it is difficult to reconcile the reported short transcription 6 factor (TF) residence times on the DNA with the high specificity of regulation. In non-equilibrium 7 models, progress is difficult due to an explosion in the number of parameters. Here, we navigate 8 this complexity by looking for minimal non-equilibrium enhancer models that yield desired regula-9 tory phenotypes: low TF residence time, high specificity and tunable cooperativity. We find that a 10 single extra parameter, interpretable as the "linking rate" by which bound TFs interact with Medi-11 ator components, enables our models to escape equilibrium bounds and access optimal regulatory 12 phenotypes, while remaining consistent with the reported phenomenology and simple enough to 13 be inferred from upcoming experiments. We further find that high specificity in non-equilibrium 14 models is in a tradeoff with gene expression noise, predicting bursty dynamics -an experimentally-15 observed hallmark of eukaryotic transcription. By drastically reducing the vast parameter space to 16 a much smaller subspace that optimally realizes biological function prior to inference from data, our 17 normative approach holds promise for mathematical models in systems biology. 18 Keywords: transcriptional regulation | non-equilibrium models | noise in gene expression | enhancer function 73 somes are assembled in equilibrium [4]. Kinetic schemes 74 may be required to match the reported characteristics of 75 bursty gene expression (vi) [26], or realize high cooper-76 ativity (vii) [27]. Thermodynamic models undisputedly 77 have statistical power to predict expression from regula-78 tory sequence even in eukaryotes [28], yet this does not 79 resolve their biophysical inconsistencies or rule out non-80 equilibrium models. Unfortunately, mechanistically de-81 tailed non-equilibrium models entail an explosion in the 82 complexity of the corresponding reaction schemes and 83 the number of associated parameters: on the one hand, 84 such models are intractable to infer from data, while on 85 2the other, it is difficult to understand which details are 86 essential for the emergence of regulatory function.
87To deal with this complexity, we systematically sim-88 plify the space of enhancer models. We adopt the norma-89 tive approach, commonly encountered in the applications 90 of optimality ideas in neuroscience and elsewhere [29-91 31]: we theoretically identify those models for which var-92 ious performance measures of gene regulation, which we 93 call "regulatory phenotypes", are maximized. Such op-94 timal model classes are our candidates that could subse-95 quently be refined for particular biological systems and 96 confronted with data. Thus, rath...