He graduated summa cum laude from Yale University in 1958 with a Bachelor of Surgery in Chemistry and earned a Bachelor of Arts from the University of Cambridge in 1960. He received his Doctor of Philosophy in Chemistry from the University of California, San Diego, in 1963, working with Bruno Zimm.
He was best known for his work on nucleic acid structure and function. Along with Victor Bloomfield, Crothers and Tinoco later co-authored two books, including the 1974 text Physical Chemistry of Nucleic Acids," updated in 2000 as "Nucleic Acids: Structures, Properties, and Functions." After a postdoctoral fellowship with Manfred Eigen at the Max-Planck-Institut in Göttingen, Germany, he returned to Yale University as a faculty member in the Department of Chemistry in 1964. He served four three-year terms as Chairman of the Yale Department of Chemistry (1975-1981.
At the time of his retirement in 2003, he was Sterling Professor of Chemistry and held a joint appointment in the Department of Molecular Biophysics and Biochemistry, of which he was a founding member. After retiring from Yale, Crothers worked as a partner and scientific consultant for numerous biotechnology firms. Crothers was a co-inventor of the Zimm–Crothers viscometer.
Among the most frequently cited papers from the Crothers lab is a method for the widely used electrophoretic mobility shift assay, better known as a "gel shift" assay, which is used to detect and estimate the affinity of protein-nucleic acid complexes.
He made many other contributions to areas including the thermodynamics of secondary structure in Ribonucleic acid and deoxyribonucleic acid, the role of magnesium in Ribonucleic acid tertiary structure, deoxyribonucleic acid-drug interactions, sequence-directed and protein-induced deoxyribonucleic acid bending, deoxyribonucleic acid cyclization kinetics, Ribonucleic acid folding in riboswitches, protein-Ribonucleic acid structure, mechanisms of transcription, and nucleosome positioning.
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Fellow American Association for the Advancement of Science, American Academy Arts and Sciences. Member National Academy of Sciences, American Society Biological Chemists, Biophysics Society (council 1979-1982).
Married Leena Kareoja, June 24, 1960. Children: Nina H., Kirstina A.