Step 1: Understanding the 'Futile Cycle'.
A futile cycle is a biochemical process where two opposing metabolic pathways run simultaneously, leading to the hydrolysis of ATP and the dissipation of energy as heat. This process typically involves reversible reactions that, when coupled, result in no net change in the products but lead to the expenditure of energy.
Step 2: Analyzing the options.
- (A) Incorrect: The reaction between glucose and glucose-6-phosphate is part of the glycolysis and gluconeogenesis pathways, but it is not part of the futile cycle, since it does not lead to energy dissipation through opposing reactions.
- (B) Correct: The interconversion between fructose-6-phosphate and fructose-1,6-bisphosphate involves the opposing actions of phosphofructokinase (PFK) and fructose-1,6-bisphosphatase, which is a classic example of a futile cycle. These opposing reactions lead to ATP hydrolysis and heat production without any net change in metabolite concentrations.
- (C) Correct: The reaction between glucose-6-phosphate and fructose-6-phosphate can also be part of a futile cycle, particularly during the process of gluconeogenesis and glycolysis, where opposing reactions involving phosphoglucoisomerase and glucose-6-phosphatase contribute to energy dissipation.
- (D) Incorrect: The reaction between 1,3-bisphosphoglycerate and 3-phosphoglycerate is involved in the glycolysis and gluconeogenesis pathways but does not typically form a futile cycle.
Final Answer:
\[
\boxed{\text{(B) and (C) are correct.}}
\]