The sequence context surrounding the AUG start codon of an open reading frame - the 'Kozak sequence' - affects the probability with which a scanning ribosome will recognize the start codon and start translating there. A significant number of transcripts in animals such as Drosophila contain weak Kozak sequences. This is predicted to cause constitutively low translation of these transcripts. We study here the additional possibility that these mRNAs have weak Kozak sequences to allow for the regulation of their translation in response to stress or altered cellular signaling. We find that transcripts with weak Kozak sequences are less sensitive to drops in global elongation rates and more sensitive to drops in global initiation rates compared to transcripts with strong Kozak sequences. This provides a mechanism by which changes in these global translation parameters differentially affect different pools of mRNAs depending on their Kozak sequence, thereby shaping the proteome. Interestingly, mRNAs with weak Kozak sequences are enriched for genes involved in neurobiology, suggesting that they constitute a functional group that can be translationally co-regulated.