= Solution
Write $c=c_1$, $\beta=(k_{-1}+k_1^+)/(k_1S_0)$ and $\kappa=k_1^+/(k_1S_0)$. Since $c_0=1-c$, the exact equations are
$$
\epsilon c_\tau=s-(s+\beta)c,\qquad s_\tau=-s+(s+\beta-\kappa)c.
$$
At the prescribed initial data $s=1,c=0$, $r_1=1$, rather than zero. The <quasi-steady-state approximation> would instead demand $c=s/(s+\beta)$, giving $c(0^+)=1/(1+\beta)$. This mismatch signals an <initial layer>.
Set $\sigma=\tau/\epsilon$ and expand $s=s_0+\epsilon s_1+\cdots$, $c=c_0+\cdots$. At leading order $s_0'=0$, so $s_0=1$, while $c_0'=1-(1+\beta)c_0$ and $c_0(0)=0$. Thus
$$
\boxed{s_{\rm in}=1+O(\epsilon),\qquad c_{\rm in}=\frac{1-e^{-(1+\beta)\sigma}}{1+\beta}+O(\epsilon).}
$$
For a more informative <substrate> approximation, integration of $s_1'=-1+(1+\beta-\kappa)c_0$ with $s_1(0)=0$ gives
$$
s_{\rm in}=1-\epsilon\left[\frac{\kappa\sigma}{1+\beta}+\frac{1+\beta-\kappa}{(1+\beta)^2}(1-e^{-(1+\beta)\sigma})\right]+\cdots.
$$
On the slow scale $c_{\rm out}=s_{\rm out}/(s_{\rm out}+\beta)$ and $s_{{\rm out},\tau}=-\kappa s_{\rm out}/(s_{\rm out}+\beta)$, with matched leading value $s_{\rm out}(0)=1$. Integrating gives $s_{\rm out}+\beta\log s_{\rm out}=1-\kappa\tau$. In the overlap $1\ll\sigma\ll1/\epsilon$, the inner complex tends to $1/(1+\beta)$ and its <substrate> slope matches the expansion of this <outer solution>. A leading <composite asymptotic expansion> is
$$
c_{\rm comp}(\tau)=\frac{s_{\rm out}(\tau)}{s_{\rm out}(\tau)+\beta}-\frac{e^{-(1+\beta)\tau/\epsilon}}{1+\beta},\qquad s_{\rm comp}=s_{\rm out}.
$$
It satisfies both initial values at leading order and smoothly joins the initial complex build-up to <substrate> depletion. The displayed $O(\epsilon)$ inner <substrate> term records the small binding-induced initial slip if greater accuracy is wanted.
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