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How does auxin help a leaning stem grow upward against gravity?

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fern life cycle — plant development The life cycle of the fern. (1) Clusters (sori) of sporangia (spore cases) grow on the undersurface of mature fern leaves. (2) Released from its spore case, the haploid spore is carried to the ground, where it germinates into a tiny, usually heart-shaped, gametophyte (gamete-producing structure), anchored to the ground by rhizoids (rootlike projections). (3) Under moist conditions, mature sperm are released from the antheridia and swim to the egg-producing archegonia that have formed on the gametophyte's lower surface. (4) When fertilization occurs, a zygote forms and develops into an embryo within the archegonium. (5) The embryo eventually grows larger than the gametophyte and becomes a sporophyte.

Auxins are plant hormones that regulate growth and development, including the response to gravity, known as gravitropism.21

When a stem is leaning, auxins accumulate along its lower side. In stems, this higher concentration of auxin stimulates the cells on the lower side to elongate more rapidly than those on the upper side. This differential growth causes the stem to bend upward, effectively returning it to a vertical position.21

This process involves the hormone acting on the chemical bonds of plant cell walls, which allows the cells to deform irreversibly as they take in water and synthesize new wall material. While this mechanism is consistent across many shoots, it is important to note that roots respond differently to the same hormone distribution, as auxins can have an inhibitory effect on root cell elongation.21