Nondeterministic Turing Machine Equivalent to Deterministic Turing Machine

Every Deterministic Turing Machine can be run by a Nondeterministic Turing Machine
The machine definitions only differ in whether $(q′,Y′,d)\in\delta(q,Y)$ or $(q′,Y′,d)=\delta(q,Y)$. Define singletons $\delta(q,Y)=\{(q',Y',d)\}$ from the deterministic mapping and run your DTM on your NTM.

Every Nondeterministic Turing Machine can be run by a Deterministic Turing Machine
Albeit slow: sequentialize the parallel search breadth-first. Define yourself a Universal Turing Machine that can single-step an instantaneous description of your NTM and follow a given NTM transition table like this: 1. for each applicable transition, copy the instantaneous description to the end of the tape and perform its step. 2. when the transition table is exhausted, mark the original instantaneous description processed and attend to the next as in 1. unless you halt for simulating an accepting state or having run out of unprocessed configurations.