DNA replication is a high-fidelity information-copying processes which is realized by DNA polymerase (DNAP). The high fidelity was explained on the basis of the well-known kinetic-proofreading mechanism (KPR), under which the so-called fidelity-speed trade-off was studied theoretically. However, numerous biochemical experiments have shown that the high fidelity of DNA replication is achieved due to the initial discrimination of polymerase domain of DNAP, as well as the proofreading of the exonuclease domain of DNAP. This exonuclease-proofreading mechanism (EPR) is totally different from KPR. So the trade-off issues are worth being re-examined under EPR. In this paper, we use the first-passage method recently proposed by us to discuss the possible trade-offs in DNA replication under EPR. We show that there could be no fidelity-speed trade-off under EPR, i.e., the fidelity and the speed can be simultaneously enhanced by EPR in a large range of kinetic parameters. This provides a new perspective to understand the experimental data of the exonuclease activity of T7 DNAP and T4 DNAP. We also show that there exists the fidelity-proofreading cost trade-off, i.e., the fidelity is enhanced at the cost of increasing the futile hydrolysis of dNTP. A possible way to avoid this trade-off is to regulate the rate of DNAP translocation: slowing down the forward translocation (in the presence of the terminal mismatch) can enhance the fidelity without changing the speed and the proofreading cost. Our theoretical analysis offers deeper insights on the kinetics-function relation of DNAP. PACS numbers: 82.39.-k, 87.15.Rn, 87.16.A-
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